This page was generated from unit-2.1.1-preconditioners/preconditioner.ipynb.

2.1.1 Preconditioners in NGSolve

Preconditioners are approximate inverses which are used within iterative methods to solve linear or non-linear equations.

Here are some built-in preconditioners in NGSolve:

• Jacobi (local) and block Jacobi

• Direct solvers using sparse factorization (direct)

• Geometric multigrid with different block-smoothers (multigrid)

• Algebraic multigrid preconditioner (h1amg)

• p-version element-level BDDC (bddc)

This tutorial quickly introduces how to use some of these within a solver, and the next two tutorials, 2.1.2 and 2.1.3, continue the preconditioner discussion.

from netgen.csg import unit_cube
from netgen.geom2d import unit_square
from ngsolve import *
import netgen.gui
import matplotlib.pyplot as plt

A simple test problem

In order to experiment with various preconditioners, let us define a simple Poisson solver with the name of a preconditioner as the keyword argument precond.

def SolveProblem(h=0.5, p=1, levels=1,
                 condense = False,
                 precond="local"):
    """
    Solve Poisson problem on l refinement levels.
        h: coarse mesh size
        p: polynomial degree
        l: number of refinement levels
        condense: if true, perform static condensations
        precond: name of a built-in preconditioner
    OUTPUT:
        List of tuples of ndofs and iterations
    """
    mesh = Mesh(unit_square.GenerateMesh(maxh=h))
    # mesh = Mesh(unit_cube.GenerateMesh(maxh=h))
    fes = H1(mesh, order=p, dirichlet="bottom|left")

    u, v = fes.TnT()
    a = BilinearForm(fes, eliminate_internal=condense)
    a += grad(u)*grad(v)*dx
    f = LinearForm(fes)
    f += v*dx
    gfu = GridFunction(fes)
    Draw(gfu)
    c = Preconditioner(a, precond) # Register c to a BEFORE assembly

    steps = []

    for l in range(levels):
        if l > 0:
            mesh.Refine()
        fes.Update()
        gfu.Update()

        with TaskManager():
            a.Assemble()
            f.Assemble()

            # Conjugate gradient solver
            inv = CGSolver(a.mat, c.mat, maxsteps=1000)

            # Solve steps depend on condense
            if condense:
                f.vec.data += a.harmonic_extension_trans * f.vec

            gfu.vec.data = inv * f.vec

            if condense:
                gfu.vec.data += a.harmonic_extension * gfu.vec
                gfu.vec.data += a.inner_solve * f.vec
        steps.append ( (fes.ndof, inv.GetSteps()) )
        if fes.ndof < 15000:
            Redraw()
    return steps

The Preconditioner registers itself to the BilinearForm. Whenever the BilinearForm is re-assembled, the Preconditioner is updated as well.

The local preconditioner

The local preconditioner is a simple Jacobi preconditioner. The number of CG-iterations with the local preconditioner is proportional to \(h^{-1} \sim 2^l\):

SolveProblem(precond="local")

 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 load internal triangle rules
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
assemble VOL element 6/6

[(8, 4)]

0 0.216506
1 0.15554
2 0.00984673
3 1.29234e-17

res_local = SolveProblem(levels=9, precond="local")
res_local

 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
assemble VOL element 6/6
0 0.216506
1 0.15554
2 0.00984673
3 1.29234e-17
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
0 0.138193
1 0.153753
2 0.116572
3 0.0713624
4 0.0324336
5 0.00752701
6 0.000971976
7 5.44153e-05
8 1.40176e-18
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
0 0.0754759
1 0.153035
2 0.0928317
3 0.0899082
4 0.102225
5 0.0723412
6 0.0472377
7 0.0424434
8 0.0246661
9 0.0118849
10 0.00490928
11 0.00214447
12 0.00103007
13 0.000393217
14 0.000135465
15 3.80968e-05
16 6.7356e-06
17 2.76331e-06
18 7.01211e-07
19 2.83454e-07
20 9.8444e-08
21 3.2501e-08
22 5.45041e-09
23 1.1504e-09
24 7.62389e-11
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
0 0.0391059
1 0.0982581
2 0.0915281
3 0.0889827
4 0.0661326
5 0.0615353
6 0.0595127
7 0.0836062
8 0.073055
9 0.0589171
10 0.0517409
11 0.0375674
12 0.0353505
13 0.0335517
14 0.0296508
15 0.0235869
16 0.0199672
17 0.0131925
18 0.00809935
19 0.00581546
20 0.00362867
21 0.0021879
22 0.00173197
23 0.00126879
24 0.00094027
25 0.000619314
26 0.000383946
27 0.00027824
28 0.000169505
29 7.05815e-05
30 3.84889e-05
31 2.34641e-05
32 1.10815e-05
33 6.67469e-06
34 3.82977e-06
35 2.45923e-06
36 1.32268e-06
37 7.53322e-07
38 5.36298e-07
39 3.98344e-07
40 2.03535e-07
41 1.06239e-07
42 8.50011e-08
43 5.20049e-08
44 2.46207e-08
45 1.23496e-08
46 6.75256e-09
47 3.13721e-09
48 1.35697e-09
49 4.81431e-10
50 2.21087e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 1536/1536
assemble VOL element 1536/1536
0 0.0198701
1 0.054748
2 0.0737023
3 0.07889
4 0.0658122
5 0.0662042
6 0.0562027
7 0.0546864
8 0.0466731
9 0.044432
10 0.0404148
11 0.0406512
12 0.0459011
13 0.0606264
14 0.056602
15 0.0484641
16 0.0520401
17 0.0474428
18 0.0417499
19 0.0399606
20 0.0325597
21 0.0303331
22 0.0252646
23 0.0257607
24 0.0253383
25 0.0258125
26 0.023731
27 0.0214748
28 0.020994
29 0.0184891
30 0.0171734
31 0.0162491
32 0.013445
33 0.0111492
34 0.0083336
35 0.00688139
36 0.00638114
37 0.0049405
38 0.00427012
39 0.0032155
40 0.00266364
41 0.00194742
42 0.00167587
43 0.00138658
44 0.00124965
45 0.00116006
46 0.000945751
47 0.000780489
48 0.000774941
49 0.000608707
50 0.000460504
51 0.000374617
52 0.000304566
53 0.000264117
54 0.000236986
55 0.000191748
56 0.000133702
57 8.68257e-05
58 6.40568e-05
59 4.98797e-05
60 3.56298e-05
61 2.78603e-05
62 2.0961e-05
63 1.46424e-05
64 1.13468e-05
65 7.4998e-06
66 5.4085e-06
67 4.36247e-06
68 3.82072e-06
69 3.16663e-06
70 2.34964e-06
71 1.88207e-06
72 1.68094e-06
73 1.12422e-06
74 8.57566e-07
75 7.25003e-07
76 6.66238e-07
77 5.90464e-07
78 5.24108e-07
79 4.00787e-07
80 2.71502e-07
81 1.77699e-07
82 1.3301e-07
83 1.01926e-07
84 8.65822e-08
85 8.27065e-08
86 6.72758e-08
87 4.78436e-08
88 3.50776e-08
89 2.34285e-08
90 1.60842e-08
91 1.31925e-08
92 9.50656e-09
93 6.71131e-09
94 5.03814e-09
95 3.28267e-09
96 2.4534e-09
97 1.6775e-09
98 1.00517e-09
99 5.74042e-10
100 3.33003e-10
101 2.29542e-10
102 1.50239e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 6144/6144
assemble VOL element 6144/6144
0 0.0100114
1 0.0288061
2 0.0552323
3 0.0611348
4 0.053092
5 0.0560453
6 0.0500889
7 0.0516498
8 0.0468878
9 0.0472492
10 0.0436619
11 0.0431667
12 0.0400934
13 0.0393489
14 0.0364938
15 0.035474
16 0.0329761
17 0.0320618
18 0.030566
19 0.0298367
20 0.0284394
21 0.0277942
22 0.0275126
23 0.0306059
24 0.0373495
25 0.0450369
26 0.0405087
27 0.0353376
28 0.0389464
29 0.0388704
30 0.0345455
31 0.0375503
32 0.0350849
33 0.0335203
34 0.0342571
35 0.0304355
36 0.0307432
37 0.0281004
38 0.0265335
39 0.0255643
40 0.0227169
41 0.0221591
42 0.0195448
43 0.0191931
44 0.0178708
45 0.0177227
46 0.0179321
47 0.0188385
48 0.0196011
49 0.0182684
50 0.017333
51 0.0165545
52 0.0174161
53 0.0158938
54 0.0153784
55 0.0147741
56 0.014272
57 0.0131673
58 0.0134569
59 0.0127177
60 0.0122569
61 0.0119786
62 0.0111072
63 0.0102743
64 0.00937333
65 0.00844958
66 0.00699895
67 0.00657921
68 0.00571792
69 0.00552987
70 0.00527966
71 0.0047675
72 0.00419341
73 0.00370033
74 0.00356779
75 0.00329654
76 0.00300836
77 0.00251758
78 0.00230048
79 0.00201143
80 0.00177731
81 0.00158091
82 0.00146886
83 0.00127326
84 0.00111053
85 0.00106321
86 0.000975953
87 0.000968049
88 0.000952674
89 0.000887585
90 0.000755773
91 0.000692812
92 0.000660015
93 0.000650943
94 0.000610185
95 0.000581688
96 0.00051965
97 0.000447091
98 0.000425045
99 0.000385102
100 0.000343888
101 0.000303976
102 0.000259706
103 0.000251674
104 0.000217229
105 0.000195074
106 0.000193079
107 0.000186907
108 0.000181702
109 0.000163736
110 0.000142539
111 0.000118047
112 8.77935e-05
113 7.43529e-05
114 6.59348e-05
115 5.80524e-05
116 4.68947e-05
117 4.47994e-05
118 3.90262e-05
119 3.08732e-05
120 2.66477e-05
121 2.39335e-05
122 2.01688e-05
123 1.85566e-05
124 1.63435e-05
125 1.36031e-05
126 1.23116e-05
127 1.0465e-05
128 8.59415e-06
129 7.09294e-06
130 5.60229e-06
131 4.95434e-06
132 3.9955e-06
133 3.77628e-06
134 3.58435e-06
135 3.24688e-06
136 3.11314e-06
137 2.52966e-06
138 2.18281e-06
139 1.89521e-06
140 1.91821e-06
141 1.80347e-06
142 1.59739e-06
143 1.45051e-06
144 1.23699e-06
145 1.06879e-06
146 8.93066e-07
147 7.85407e-07
148 7.01707e-07
149 6.14045e-07
150 5.84985e-07
151 5.49276e-07
152 5.10322e-07
153 4.86333e-07
154 4.91112e-07
155 4.58508e-07
156 4.35477e-07
157 3.75315e-07
158 3.13586e-07
159 2.74946e-07
160 2.25907e-07
161 1.85346e-07
162 1.54166e-07
163 1.31353e-07
164 1.09474e-07
165 9.33418e-08
166 8.23605e-08
167 7.46061e-08
168 6.82702e-08
169 6.46301e-08
170 6.17579e-08
171 5.8257e-08
172 5.49783e-08
173 4.99157e-08
174 4.09175e-08
175 3.52674e-08
176 3.0255e-08
177 2.45292e-08
178 2.10518e-08
179 1.81872e-08
180 1.5257e-08
181 1.2682e-08
182 1.12481e-08
183 9.92463e-09
184 8.61583e-09
185 7.16803e-09
186 6.10972e-09
187 4.95122e-09
188 4.18979e-09
189 3.51085e-09
190 2.97932e-09
191 2.54934e-09
192 2.35395e-09
193 2.03706e-09
194 1.57281e-09
195 1.21004e-09
196 9.23589e-10
197 7.55901e-10
198 6.01313e-10
199 4.76445e-10
200 3.6807e-10
201 2.80769e-10
202 2.33644e-10
203 1.97595e-10
204 1.6403e-10
205 1.33325e-10
206 1.04342e-10
207 8.07123e-11
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24576/24576
assemble VOL element 24576/24576
0 0.00502445
1 0.0147647
2 0.0401499
3 0.0450703
4 0.039829
5 0.0428882
6 0.0390656
7 0.0410851
8 0.0381025
9 0.0393504
10 0.0370909
11 0.0377645
12 0.035907
13 0.0363288
14 0.0346729
15 0.0348832
16 0.0334612
17 0.0334525
18 0.0322246
19 0.0320532
20 0.0309705
21 0.0306833
22 0.0296818
23 0.0293567
24 0.0283752
25 0.0280132
26 0.0271187
27 0.0266309
28 0.0258632
29 0.0253082
30 0.0245366
31 0.0240271
32 0.0232335
33 0.0228394
34 0.0223893
35 0.0220547
36 0.0216308
37 0.021278
38 0.0208505
39 0.0205178
40 0.0200659
41 0.0197451
42 0.0193254
43 0.0190552
44 0.0190737
45 0.019943
46 0.0225919
47 0.0274715
48 0.0321017
49 0.0311461
50 0.0265201
51 0.0261729
52 0.0292523
53 0.0280916
54 0.025554
55 0.0274987
56 0.0276628
57 0.025291
58 0.0268718
59 0.0266028
60 0.0249219
61 0.0264747
62 0.025167
63 0.02465
64 0.0256366
65 0.0238007
66 0.0245598
67 0.0236908
68 0.0227653
69 0.0232186
70 0.0215669
71 0.0219211
72 0.0208605
73 0.0202985
74 0.0201732
75 0.0188495
76 0.0191312
77 0.0177583
78 0.0177308
79 0.0167973
80 0.0162041
81 0.0157244
82 0.0147893
83 0.0146734
84 0.0137611
85 0.0137749
86 0.0130651
87 0.0129189
88 0.0124948
89 0.0122743
90 0.0123785
91 0.0124508
92 0.0134378
93 0.0138939
94 0.0146569
95 0.0136525
96 0.0129665
97 0.0121499
98 0.012706
99 0.012835
100 0.0124692
101 0.0119268
102 0.0120568
103 0.0122307
104 0.0115655
105 0.0114703
106 0.0113781
107 0.0112123
108 0.0105776
109 0.0108007
110 0.0102644
111 0.0100021
112 0.00989042
113 0.00977779
114 0.00939894
115 0.00956251
116 0.00924536
117 0.00904243
118 0.00913524
119 0.00875021
120 0.00870519
121 0.00857344
122 0.00828988
123 0.00808451
124 0.00791059
125 0.00736355
126 0.00729003
127 0.0067073
128 0.0064953
129 0.00628979
130 0.00547779
131 0.00557218
132 0.00485599
133 0.00452225
134 0.00451467
135 0.00408072
136 0.00431016
137 0.00403923
138 0.00425822
139 0.00395646
140 0.00376109
141 0.00328684
142 0.00299246
143 0.00283908
144 0.00296636
145 0.00293849
146 0.00280562
147 0.00248923
148 0.00249783
149 0.00243007
150 0.00233307
151 0.00210594
152 0.00198969
153 0.00198425
154 0.00180222
155 0.0016609
156 0.00156445
157 0.00150604
158 0.00135964
159 0.00130666
160 0.0012817
161 0.00118997
162 0.00111275
163 0.00106263
164 0.000998686
165 0.000927449
166 0.000889567
167 0.000834997
168 0.000778864
169 0.000775051
170 0.000725713
171 0.000713683
172 0.00070274
173 0.00069071
174 0.000685839
175 0.000720607
176 0.000681865
177 0.000636436
178 0.000574931
179 0.000541658
180 0.000542112
181 0.000504956
182 0.000484294
183 0.00049361
184 0.00047335
185 0.000459158
186 0.000470632
187 0.00044537
188 0.000430842
189 0.000419945
190 0.000388896
191 0.000390262
192 0.000361704
193 0.000337997
194 0.000322791
195 0.000311795
196 0.000302002
197 0.000284568
198 0.000263641
199 0.000253574
200 0.000239455
201 0.000227087
202 0.000213034
203 0.000205288
204 0.000197935
205 0.000179626
206 0.000166105
207 0.000154578
208 0.00015143
209 0.000144503
210 0.000143634
211 0.000142381
212 0.00013758
213 0.000134945
214 0.000133163
215 0.00013097
216 0.000127861
217 0.000124916
218 0.000123158
219 0.000110239
220 9.96772e-05
221 8.6462e-05
222 7.49498e-05
223 6.89942e-05
224 6.5778e-05
225 6.15166e-05
226 5.21503e-05
227 4.69393e-05
228 4.48478e-05
229 4.41474e-05
230 4.10623e-05
231 3.70411e-05
232 3.68151e-05
233 3.40892e-05
234 3.08591e-05
235 2.82161e-05
236 2.73794e-05
237 2.39389e-05
238 2.21533e-05
239 2.09227e-05
240 1.9038e-05
241 1.8001e-05
242 1.73446e-05
243 1.56741e-05
244 1.46922e-05
245 1.39225e-05
246 1.31091e-05
247 1.24632e-05
248 1.17949e-05
249 1.06258e-05
250 1.03268e-05
251 9.40226e-06
252 8.68605e-06
253 8.07592e-06
254 7.34118e-06
255 6.88419e-06
256 6.12379e-06
257 5.46663e-06
258 5.0344e-06
259 4.52955e-06
260 4.00314e-06
261 3.81534e-06
262 3.35863e-06
263 3.20347e-06
264 2.95258e-06
265 2.81237e-06
266 2.7404e-06
267 2.60901e-06
268 2.69359e-06
269 2.53381e-06
270 2.49433e-06
271 2.24824e-06
272 2.13192e-06
273 1.86765e-06
274 1.72861e-06
275 1.62819e-06
276 1.5579e-06
277 1.56243e-06
278 1.48099e-06
279 1.40704e-06
280 1.36142e-06
281 1.33713e-06
282 1.30989e-06
283 1.23361e-06
284 1.20526e-06
285 1.12851e-06
286 1.05555e-06
287 9.74662e-07
288 9.07161e-07
289 8.48884e-07
290 7.73129e-07
291 7.08072e-07
292 6.64193e-07
293 6.0985e-07
294 5.68542e-07
295 5.33048e-07
296 5.04973e-07
297 4.84574e-07
298 4.74243e-07
299 4.465e-07
300 4.32231e-07
301 3.99838e-07
302 3.81433e-07
303 3.72853e-07
304 3.71781e-07
305 3.72672e-07
306 3.64189e-07
307 3.5375e-07
308 3.46306e-07
309 3.44641e-07
310 3.30461e-07
311 3.32743e-07
312 3.18774e-07
313 2.98029e-07
314 2.77163e-07
315 2.56003e-07
316 2.32858e-07
317 2.08334e-07
318 1.9036e-07
319 1.79837e-07
320 1.69629e-07
321 1.52735e-07
322 1.39712e-07
323 1.22574e-07
324 1.14259e-07
325 1.04795e-07
326 9.81964e-08
327 8.89078e-08
328 7.98572e-08
329 7.43139e-08
330 6.93136e-08
331 6.4026e-08
332 6.08714e-08
333 5.68861e-08
334 5.53927e-08
335 5.22267e-08
336 4.91625e-08
337 4.74136e-08
338 4.49527e-08
339 4.51224e-08
340 4.42961e-08
341 4.45076e-08
342 4.22205e-08
343 4.25207e-08
344 4.10867e-08
345 3.91224e-08
346 3.62715e-08
347 3.27377e-08
348 2.99319e-08
349 2.80024e-08
350 2.57795e-08
351 2.40535e-08
352 2.21733e-08
353 2.0393e-08
354 1.88913e-08
355 1.74661e-08
356 1.5914e-08
357 1.45429e-08
358 1.35837e-08
359 1.23352e-08
360 1.13271e-08
361 1.04405e-08
362 9.66838e-09
363 8.97818e-09
364 8.42244e-09
365 8.18839e-09
366 7.6481e-09
367 7.06559e-09
368 6.32378e-09
369 5.67688e-09
370 5.29031e-09
371 5.12171e-09
372 4.78583e-09
373 4.22353e-09
374 3.7481e-09
375 3.43033e-09
376 3.21168e-09
377 2.87007e-09
378 2.5392e-09
379 2.3876e-09
380 2.23556e-09
381 2.16395e-09
382 2.03543e-09
383 1.92703e-09
384 1.83052e-09
385 1.67582e-09
386 1.47419e-09
387 1.32289e-09
388 1.19743e-09
389 1.07981e-09
390 9.16081e-10
391 8.34195e-10
392 7.40484e-10
393 6.6755e-10
394 6.0332e-10
395 5.26899e-10
396 4.71207e-10
397 4.14538e-10
398 3.69381e-10
399 3.15539e-10
400 2.8307e-10
401 2.47414e-10
402 2.23325e-10
403 2.012e-10
404 1.91358e-10
405 1.79062e-10
406 1.6447e-10
407 1.48482e-10
408 1.31456e-10
409 1.16192e-10
410 1.04441e-10
411 9.57466e-11
412 8.35499e-11
413 7.39757e-11
414 6.68971e-11
415 5.90155e-11
416 5.10083e-11
417 4.59855e-11
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 98304/98304
assemble VOL element 98304/98304
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1 0.00747323
2 0.0287762
3 0.0325119
4 0.0289678
5 0.03148
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447 4.58369e-05
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494 8.67255e-06
495 8.59001e-06
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500 7.11745e-06
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502 6.56994e-06
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506 5.766e-06
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510 4.68902e-06
511 4.54701e-06
512 4.35533e-06
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514 3.91366e-06
515 3.64395e-06
516 3.41723e-06
517 3.38693e-06
518 3.10789e-06
519 3.02936e-06
520 2.85534e-06
521 2.72489e-06
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523 2.47354e-06
524 2.4329e-06
525 2.26285e-06
526 2.26634e-06
527 2.12189e-06
528 2.11573e-06
529 2.02604e-06
530 2.01599e-06
531 1.96797e-06
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535 1.89911e-06
536 1.88043e-06
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556 1.02583e-06
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560 9.81133e-07
561 9.63493e-07
562 9.46377e-07
563 9.38488e-07
564 9.25652e-07
565 8.98032e-07
566 8.93219e-07
567 8.56597e-07
568 8.51528e-07
569 8.13567e-07
570 7.99667e-07
571 7.75874e-07
572 7.49767e-07
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574 6.97344e-07
575 6.77399e-07
576 6.45207e-07
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580 5.48516e-07
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582 5.079e-07
583 4.84004e-07
584 4.66253e-07
585 4.49263e-07
586 4.31064e-07
587 4.13948e-07
588 3.99045e-07
589 3.92059e-07
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591 3.72984e-07
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593 3.59191e-07
594 3.53306e-07
595 3.41749e-07
596 3.38449e-07
597 3.24161e-07
598 3.18498e-07
599 3.04505e-07
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601 2.84004e-07
602 2.80345e-07
603 2.74716e-07
604 2.74614e-07
605 2.78351e-07
606 2.75194e-07
607 2.76316e-07
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611 2.58831e-07
612 2.53098e-07
613 2.50357e-07
614 2.49298e-07
615 2.52169e-07
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617 2.46276e-07
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619 2.42127e-07
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621 2.36153e-07
622 2.33964e-07
623 2.29235e-07
624 2.27727e-07
625 2.21025e-07
626 2.17976e-07
627 2.07543e-07
628 1.97655e-07
629 1.84793e-07
630 1.71778e-07
631 1.63352e-07
632 1.59452e-07
633 1.57019e-07
634 1.53187e-07
635 1.49295e-07
636 1.41878e-07
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640 1.16656e-07
641 1.10891e-07
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643 1.01818e-07
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645 9.36848e-08
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651 7.10845e-08
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655 5.99595e-08
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657 5.49061e-08
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660 4.90663e-08
661 4.76156e-08
662 4.62348e-08
663 4.43447e-08
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665 4.23657e-08
666 4.10967e-08
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668 3.84886e-08
669 3.72778e-08
670 3.59866e-08
671 3.51566e-08
672 3.43458e-08
673 3.40824e-08
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676 3.28468e-08
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681 3.15664e-08
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700 1.83115e-08
701 1.75272e-08
702 1.71553e-08
703 1.67524e-08
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705 1.53795e-08
706 1.4569e-08
707 1.40507e-08
708 1.35344e-08
709 1.28713e-08
710 1.21278e-08
711 1.17495e-08
712 1.12467e-08
713 1.07171e-08
714 1.03302e-08
715 1.01133e-08
716 9.94028e-09
717 9.57737e-09
718 8.99567e-09
719 8.58061e-09
720 7.91885e-09
721 7.2964e-09
722 6.93808e-09
723 6.8928e-09
724 6.82426e-09
725 6.86995e-09
726 6.87207e-09
727 6.63772e-09
728 6.32026e-09
729 6.02071e-09
730 5.68153e-09
731 5.40388e-09
732 5.19075e-09
733 5.01026e-09
734 4.9538e-09
735 4.79315e-09
736 4.596e-09
737 4.34302e-09
738 4.11147e-09
739 3.97663e-09
740 3.94645e-09
741 3.79723e-09
742 3.6472e-09
743 3.43141e-09
744 3.30337e-09
745 3.14266e-09
746 2.99717e-09
747 2.85784e-09
748 2.72629e-09
749 2.61358e-09
750 2.50091e-09
751 2.34365e-09
752 2.21748e-09
753 2.12717e-09
754 2.00619e-09
755 1.9185e-09
756 1.81588e-09
757 1.74803e-09
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759 1.64119e-09
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761 1.61866e-09
762 1.58385e-09
763 1.5555e-09
764 1.49745e-09
765 1.43919e-09
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769 1.25852e-09
770 1.1888e-09
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772 1.06468e-09
773 9.94929e-10
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807 1.49248e-10
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810 1.27627e-10
811 1.22432e-10
812 1.17388e-10
813 1.13944e-10
814 1.07536e-10
815 1.00566e-10
816 9.33896e-11
817 8.67854e-11
818 8.28071e-11
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820 7.69677e-11
821 7.3602e-11
822 6.96764e-11
823 6.53671e-11
824 6.10011e-11
825 5.7444e-11
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827 4.96162e-11
828 4.61527e-11
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830 4.17093e-11
831 3.9185e-11
832 3.65294e-11
833 3.44358e-11
834 3.32725e-11
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836 3.13729e-11
837 2.97347e-11
838 2.82153e-11
839 2.73939e-11
840 2.62258e-11
841 2.4471e-11
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 393216/393216
assemble VOL element 393216/393216
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925 1.85787e-05
926 1.7982e-05
927 1.82215e-05
928 1.75779e-05
929 1.70781e-05
930 1.73028e-05
931 1.71859e-05
932 1.65497e-05
933 1.63258e-05
934 1.63867e-05
935 1.57198e-05
936 1.54027e-05
937 1.5458e-05
938 1.50868e-05
939 1.44544e-05
940 1.41792e-05
941 1.3998e-05
942 1.31969e-05
943 1.31552e-05
944 1.30891e-05
945 1.24598e-05
946 1.22969e-05
947 1.22463e-05
948 1.18297e-05
949 1.15085e-05
950 1.16887e-05
951 1.10481e-05
952 1.09829e-05
953 1.09472e-05
954 1.07151e-05
955 1.04589e-05
956 1.04368e-05
957 1.00214e-05
958 9.81706e-06
959 9.89198e-06
960 9.62512e-06
961 9.39397e-06
962 9.359e-06
963 9.04771e-06
964 8.97698e-06
965 8.88118e-06
966 8.66111e-06
967 8.40771e-06
968 8.46083e-06
969 8.19473e-06
970 8.0398e-06
971 8.0175e-06
972 7.7251e-06
973 7.69991e-06
974 7.57191e-06
975 7.41354e-06
976 7.31305e-06
977 7.25071e-06
978 7.05139e-06
979 7.05558e-06
980 6.9164e-06
981 6.74647e-06
982 6.84222e-06
983 6.58029e-06
984 6.58885e-06
985 6.4898e-06
986 6.369e-06
987 6.30425e-06
988 6.2179e-06
989 6.03919e-06
990 6.02534e-06
991 5.8747e-06
992 5.73783e-06
993 5.69562e-06
994 5.54641e-06
995 5.48264e-06
996 5.41499e-06
997 5.23878e-06
998 5.25234e-06
999 5.06308e-06

[(8, 4),
 (21, 9),
 (65, 25),
 (225, 51),
 (833, 103),
 (3201, 208),
 (12545, 418),
 (49665, 842),
 (197633, 1001)]

1000 5.01013e-06

Multigrid preconditioner

A geometric multigrid Preconditioner uses the sequence of refined meshes to define a preconditioner yielding optimal iteration numbers (and complexity). It uses a direct solve on the coarsest level, and block Gauss-Seidel smoothers on the refined levels.

res_mg = SolveProblem(levels=9, precond="multigrid")
res_mg

 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.328876
1 2.04668e-17
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.356848
1 0.00912278
2 8.39618e-05
3 2.12427e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.35568
1 0.0107157
2 0.000587028
3 2.70143e-05
4 6.53175e-07
5 2.35622e-08
6 5.52189e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35423
1 0.0116601
2 0.000744241
3 3.8359e-05
4 1.65924e-06
5 1.17605e-07
6 6.9037e-09
7 4.84505e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 1536/1536
assemble VOL element 1536/1536
BlockJacobi Preconditioner, constructor called, #blocks = 2369
BlockJacobi Preconditioner built
assemble VOL element 1536/1536
0 0.353707
1 0.0126748
2 0.000811767
3 5.97099e-05
4 4.06647e-06
5 2.61384e-07
6 1.88255e-08
7 1.63407e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 6144/6144
assemble VOL element 6144/6144
BlockJacobi Preconditioner, constructor called, #blocks = 9345
BlockJacobi Preconditioner built
assemble VOL element 6144/6144
0 0.353558
1 0.0130171
2 0.000847326
3 5.84688e-05
4 5.12656e-06
5 4.08975e-07
6 2.52073e-08
7 1.89461e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24576/24576
assemble VOL element 24576/24576
BlockJacobi Preconditioner, constructor called, #blocks = 37121
BlockJacobi Preconditioner built
assemble VOL element 24576/24576
0 0.353519
1 0.0131081
2 0.00087601
3 5.64232e-05
4 4.93073e-06
5 4.40825e-07
6 3.40421e-08
7 2.32512e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 98304/98304
assemble VOL element 98304/98304
BlockJacobi Preconditioner, constructor called, #blocks = 147969
BlockJacobi Preconditioner built
assemble VOL element 98304/98304
0 0.353509
1 0.013131
2 0.00088764
3 5.76141e-05
4 4.455e-06
5 4.06586e-07
6 3.58684e-08
7 2.89833e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 393216/393216
assemble VOL element 393216/393216
BlockJacobi Preconditioner, constructor called, #blocks = 590849
BlockJacobi Preconditioner built
assemble VOL element 393216/393216
0 0.353507
1 0.0131367
2 0.000891195
3 5.86915e-05
4 4.36277e-06
5 3.47541e-07
6 3.52863e-08
7 3.08244e-09

[(8, 2),
 (21, 4),
 (65, 7),
 (225, 8),
 (833, 8),
 (3201, 8),
 (12545, 8),
 (49665, 8),
 (197633, 8)]

plt.xscale("log")
plt.yscale("log")
plt.plot(*zip(*res_local), "-*")
plt.plot(*zip(*res_mg), "-+")
plt.legend(['local', 'mg'])
plt.show()

Multigrid implementation for higher order spaces

For high order finite element spaces we use hierarchical bases. Let the (small) sub-spaces \(V_E\), \(V_F\), and \(V_C\) be those generated by basis functions associated with an edge \(E\), a face \(F\), and a cell \(C\), respectively. Then the full space admits the decomposition

\[V_{hp} = V_{p=1} + \sum_{\text{edges }E} V_E + \sum_{\text{faces }F} V_F + \sum_{\text{cells }C} V_C\]

where \(V_{p=1}\) refers to the lowest order finite element subspace. The system matrix then takes the block form

\[\begin{split}A = \left( \begin{array}{cccc} A_{VV} & A_{VE} & A_{VF} & A_{VC} \\ A_{EV} & A_{EE} & A_{EF} & A_{EC} \\ A_{FV} & A_{FE} & A_{FF} & A_{FC} \\ A_{CV} & A_{CE} & A_{CF} & A_{CC} \\ \end{array} \right)\end{split}\]

where the \(A_{VV}\)-block is exactly the system matrix of a lowest order method, representing the part of operator acting on the \(V_{p=1}\) subspace.

NGSolve’s multigrid implementation for a high order method uses h-version multigrid for the lowest order block, and local block-smoothing for the high order bubble basis functions.

for p in range(1,10):
    r = SolveProblem(h=0.5, p=p, levels=4, condense=False,
                     precond="multigrid")
    print ("p=", p, ": ndof,nsteps=", r)

 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.328876
1 2.04668e-17
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.356848
1 0.00912278
2 8.39618e-05
3 2.12427e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.35568
1 0.0107157
2 0.000587028
3 2.70143e-05
4 6.53175e-07
5 2.35622e-08
6 5.52189e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35423
1 0.0116601
2 0.000744241
3 3.8359e-05
4 1.65924e-06
5 1.17605e-07
6 6.9037e-09
p=7 4.84505e-10
 1 : ndof,nsteps= [(8, 2), (21, 4), (65, 7), (225, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.361394
1 0.0124871
2 0.000100053
3 3.66458e-07
4 1.90233e-20
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361122
1 0.0117365
2 0.000512379
3 2.37102e-05
4 8.62628e-07
5 1.77994e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.35605
1 0.0119747
2 0.000709088
3 3.47905e-05
4 2.01234e-06
5 1.26119e-07
6 7.23945e-09
7 2.83652e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35426
1 0.0120479
2 0.000804714
3 5.21269e-05
4 2.59939e-06
5 1.85087e-07
6 1.16487e-08
p=7 8.25998e-10
 2 : ndof,nsteps= [(21, 5), (65, 6), (225, 8), (833, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.359456
1 0.0244755
2 0.00614803
3 0.000996401
4 7.38031e-05
5 8.70631e-06
6 2.47374e-07
7 2.83633e-08
8 5.72124e-10
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361954
1 0.0133697
2 0.00258368
3 0.000852135
4 0.000117471
5 2.16679e-05
6 4.60798e-06
7 1.25319e-06
8 1.76905e-07
9 3.48411e-08
10 7.18174e-09
11 9.72783e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.356212
1 0.0133352
2 0.0025404
3 0.000800241
4 0.000171852
5 3.28894e-05
6 8.50707e-06
7 1.82736e-06
8 3.42158e-07
9 6.75265e-08
10 1.25258e-08
11 2.58619e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.354315
1 0.0122965
2 0.00166196
3 0.000670941
4 0.000126275
5 2.49854e-05
6 6.12428e-06
7 1.48112e-06
8 3.00089e-07
9 7.78086e-08
10 1.70598e-08
11 3.92079e-09
12 9.7946e-10
p= 3 : ndof,nsteps= [(40, 9), (133, 12), (481, 12), (1825, 13)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.359969
1 0.0234494
2 0.0086089
3 0.00338464
4 0.000481116
5 4.60723e-05
6 1.4908e-05
7 2.02488e-06
8 2.70377e-07
9 3.27915e-08
10 6.57947e-09
11 6.88784e-10
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361968
1 0.0131818
2 0.00237632
3 0.0013302
4 0.00049779
5 0.000125239
6 4.791e-05
7 1.22343e-05
8 3.24249e-06
9 1.31485e-06
10 3.65038e-07
11 1.14982e-07
12 3.07565e-08
13 8.9457e-09
14 1.77139e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.35621
1 0.0128598
2 0.00255163
3 0.00123847
4 0.000547479
5 0.000113633
6 4.40637e-05
7 1.76221e-05
8 4.79502e-06
9 1.72078e-06
10 5.218e-07
11 1.61305e-07
12 4.92828e-08
13 1.81738e-08
14 5.34336e-09
15 2.07611e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.354318
1 0.0121419
2 0.0015201
3 0.000925078
4 0.000359074
5 9.32902e-05
6 3.13138e-05
7 1.2438e-05
8 4.218e-06
9 1.47534e-06
10 5.28822e-07
11 2.169e-07
12 6.6258e-08
13 2.55985e-08
14 9.05164e-09
p=15 3.33145e-09
 4 : ndof,nsteps= [(65, 12), (225, 15), (833, 16), (3201, 16)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.359957
1 0.0222058
2 0.00748831
3 0.00401941
4 0.0018542
5 0.000377662
6 0.000110158
7 4.68164e-05
8 1.21244e-05
9 2.20442e-06
10 2.24247e-07
11 3.82368e-08
12 1.1457e-08
13 2.13243e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361972
1 0.0132969
2 0.0021435
3 0.00117303
4 0.000791478
5 0.000278533
6 0.000101052
7 5.13397e-05
8 2.32709e-05
9 8.66493e-06
10 3.45591e-06
11 1.75787e-06
12 7.15582e-07
13 2.97301e-07
14 1.52622e-07
15 5.17889e-08
16 2.29292e-08
17 7.5598e-09
18 3.33997e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.356215
1 0.0129091
2 0.00236979
3 0.00114972
4 0.000736824
5 0.000311529
6 9.21276e-05
7 4.06501e-05
8 2.18824e-05
9 9.25056e-06
10 3.68202e-06
11 1.66316e-06
12 7.16013e-07
13 2.76203e-07
14 1.20906e-07
15 5.58764e-08
16 2.00808e-08
17 9.87819e-09
18 4.55085e-09
19 1.74943e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.354319
1 0.0121757
2 0.00140658
3 0.000755175
4 0.000512128
5 0.000216501
6 7.0622e-05
7 2.81154e-05
8 1.71452e-05
9 6.34236e-06
10 2.92901e-06
11 1.2884e-06
12 5.81417e-07
13 2.67604e-07
14 1.36002e-07
15 5.83572e-08
16 2.37509e-08
17 1.21645e-08
18 5.17328e-09
p=19 2.1847e-09
 5 : ndof,nsteps= [(96, 14), (341, 19), (1281, 20), (4961, 20)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.359972
1 0.0224566
2 0.00720976
3 0.00341932
4 0.00226685
5 0.00112015
6 0.00030411
7 0.000148789
8 7.28847e-05
9 1.91382e-05
10 4.87475e-06
11 6.51662e-07
12 1.52008e-07
13 5.38682e-08
14 1.21243e-08
15 1.74456e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361972
1 0.0133186
2 0.00217314
3 0.000992665
4 0.000737282
5 0.000439782
6 0.000163087
7 8.13532e-05
8 4.77515e-05
9 2.53869e-05
10 1.0426e-05
11 4.5657e-06
12 3.28822e-06
13 1.51831e-06
14 6.42007e-07
15 2.79159e-07
16 1.46146e-07
17 9.85425e-08
18 4.66259e-08
19 2.28147e-08
20 7.04309e-09
21 3.97408e-09
22 2.29866e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.356217
1 0.0129011
2 0.00235801
3 0.00105812
4 0.000649316
5 0.000470673
6 0.000202317
7 6.80302e-05
8 3.16711e-05
9 2.26373e-05
10 1.02522e-05
11 4.11001e-06
12 2.29471e-06
13 1.35232e-06
14 6.44541e-07
15 2.7266e-07
16 1.29042e-07
17 7.58445e-08
18 3.39706e-08
19 1.7597e-08
20 9.31434e-09
21 4.96264e-09
22 2.4896e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35432
1 0.0121755
2 0.00143197
3 0.000666003
4 0.000430813
5 0.000331213
6 0.000126834
7 4.88685e-05
8 2.25863e-05
9 1.86529e-05
10 6.37545e-06
11 2.89807e-06
12 1.71535e-06
13 9.59764e-07
14 4.63665e-07
15 2.20369e-07
16 1.28607e-07
17 6.674e-08
18 3.69246e-08
19 1.93528e-08
20 1.01875e-08
21 4.91375e-09
p=22 2.29135e-09
 6 : ndof,nsteps= [(133, 16), (481, 23), (1825, 23), (7105, 23)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.359976
1 0.0224135
2 0.00726305
3 0.00296477
4 0.00219982
5 0.00136847
6 0.00063273
7 0.000240728
8 0.000140377
9 8.04578e-05
10 1.67709e-05
11 6.27141e-06
12 1.58217e-06
13 3.08362e-07
14 5.80586e-08
15 2.09396e-08
16 7.46372e-09
17 1.68248e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361973
1 0.0133193
2 0.00220751
3 0.000955483
4 0.00059175
5 0.000484005
6 0.000254144
7 0.000103897
8 6.41615e-05
9 4.07649e-05
10 2.32398e-05
11 1.05487e-05
12 4.74696e-06
13 3.12335e-06
14 2.33411e-06
15 1.10546e-06
16 4.88482e-07
17 2.33029e-07
18 1.17688e-07
19 7.67847e-08
20 4.76627e-08
21 3.40626e-08
22 1.25684e-08
23 4.75293e-09
24 3.13464e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.356217
1 0.012892
2 0.00235162
3 0.0010574
4 0.000559182
5 0.000459251
6 0.000288241
7 0.000139033
8 4.56938e-05
9 2.55859e-05
10 2.09996e-05
11 1.09511e-05
12 4.61845e-06
13 2.23842e-06
14 1.57762e-06
15 9.76281e-07
16 5.05567e-07
17 2.3891e-07
18 1.28847e-07
19 7.61114e-08
20 4.01641e-08
21 2.27956e-08
22 1.3198e-08
23 6.6357e-09
24 4.60974e-09
25 2.61925e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35432
1 0.0121731
2 0.00144442
3 0.000678642
4 0.00034123
5 0.000336768
6 0.000193311
7 7.8881e-05
8 3.39952e-05
9 1.8475e-05
10 1.58817e-05
11 7.6655e-06
12 2.78043e-06
13 1.57371e-06
14 1.12571e-06
15 6.5602e-07
16 3.27258e-07
17 1.72413e-07
18 1.08939e-07
19 6.12785e-08
20 3.40735e-08
21 2.18247e-08
22 1.26022e-08
23 7.4786e-09
24 4.18833e-09
p=25 1.93666e-09
 7 : ndof,nsteps= [(176, 18), (645, 25), (2465, 26), (9633, 26)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.359978
1 0.0224181
2 0.0073094
3 0.00295162
4 0.00179397
5 0.00150433
6 0.000819236
7 0.000402255
8 0.000197161
9 0.000125012
10 6.66218e-05
11 1.3268e-05
12 6.14669e-06
13 1.90745e-06
14 4.44854e-07
15 7.90921e-08
16 2.05004e-08
17 8.23094e-09
18 2.94443e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361973
1 0.0133191
2 0.00221793
3 0.000970903
4 0.000533417
5 0.000402827
6 0.000340919
7 0.000141759
8 7.12831e-05
9 5.19367e-05
10 3.43658e-05
11 2.01144e-05
12 9.84564e-06
13 4.28591e-06
14 2.59446e-06
15 2.02751e-06
16 1.49712e-06
17 6.74026e-07
18 3.69433e-07
19 1.82975e-07
20 8.99757e-08
21 5.29832e-08
22 3.47831e-08
23 2.95102e-08
24 1.69971e-08
25 5.91171e-09
26 2.96804e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.356218
1 0.012891
2 0.00232895
3 0.00108079
4 0.000532419
5 0.000376933
6 0.000342118
7 0.000187856
8 9.17848e-05
9 3.36332e-05
10 2.0724e-05
11 1.8386e-05
12 1.05698e-05
13 5.06466e-06
14 2.10686e-06
15 1.37865e-06
16 1.08995e-06
17 6.87552e-07
18 3.60477e-07
19 1.86259e-07
20 1.03429e-07
21 5.7282e-08
22 3.65705e-08
23 2.23612e-08
24 1.58692e-08
25 7.67513e-09
26 4.32134e-09
27 3.057e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35432
1 0.0121727
2 0.00144005
3 0.000700948
4 0.000317699
5 0.000256249
6 0.000250342
7 0.00010683
8 5.53197e-05
9 2.51694e-05
10 1.51441e-05
11 1.33808e-05
12 8.11248e-06
13 3.22234e-06
14 1.28802e-06
15 1.01662e-06
16 7.26614e-07
17 4.43309e-07
18 2.29878e-07
19 1.25474e-07
20 7.60395e-08
21 4.58657e-08
22 3.02866e-08
23 2.0897e-08
24 1.3451e-08
25 6.8126e-09
26 4.20302e-09
p=27 2.84639e-09
 8 : ndof,nsteps= [(225, 19), (833, 27), (3201, 28), (12545, 28)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 14
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.359978
1 0.0224286
2 0.00728216
3 0.0030513
4 0.00156596
5 0.00135403
6 0.00101342
7 0.000494472
8 0.000275257
9 0.000150186
10 0.000110912
11 5.49871e-05
12 1.26439e-05
13 4.32293e-06
14 1.81698e-06
15 7.45218e-07
16 1.58682e-07
17 3.534e-08
18 8.46984e-09
19 2.8639e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 45
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361973
1 0.0133191
2 0.00222004
3 0.00098899
4 0.000524108
5 0.000338334
6 0.000322627
7 0.000223234
8 8.42403e-05
9 5.28708e-05
10 4.23828e-05
11 2.79507e-05
12 1.72283e-05
13 8.91821e-06
14 3.51449e-06
15 2.37705e-06
16 1.6091e-06
17 1.49956e-06
18 8.58668e-07
19 4.5347e-07
20 2.79885e-07
21 1.43475e-07
22 6.21495e-08
23 3.41145e-08
24 2.46548e-08
25 2.16191e-08
26 1.56952e-08
27 6.76776e-09
28 2.79095e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 161
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.356218
1 0.0128928
2 0.00231065
3 0.00109012
4 0.000543228
5 0.000328532
6 0.000308634
7 0.000231532
8 0.000128087
9 5.93822e-05
10 2.60538e-05
11 1.66575e-05
12 1.59393e-05
13 1.07274e-05
14 5.34874e-06
15 2.16139e-06
16 1.29307e-06
17 9.91126e-07
18 8.08408e-07
19 4.77056e-07
20 2.63205e-07
21 1.5127e-07
22 9.11004e-08
23 4.75745e-08
24 3.21976e-08
25 2.11827e-08
26 1.64081e-08
27 9.18541e-09
28 4.36814e-09
29 2.81263e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 609
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35432
1 0.0121729
2 0.00143463
3 0.000709017
4 0.000328269
5 0.000211394
6 0.000214891
7 0.000160378
8 6.80673e-05
9 3.98856e-05
10 1.82794e-05
11 1.26107e-05
12 1.14938e-05
13 7.83493e-06
14 3.67854e-06
15 1.29178e-06
16 8.76196e-07
17 6.79931e-07
18 5.11356e-07
19 2.9506e-07
20 1.87221e-07
21 1.08198e-07
22 6.28609e-08
23 3.23123e-08
24 2.55032e-08
25 1.90421e-08
26 1.39291e-08
27 6.54876e-09
28 3.74174e-09
p=29 2.85318e-09
 9 : ndof,nsteps= [(280, 20), (1045, 29), (4033, 30), (15841, 30)]

We observe that the number of iterations grows mildly with the order, and remains bounded with mesh refinement.

Performing static condensation improves the situation:

for p in range(1,10):
    r = SolveProblem(h=0.5, p=p, levels=4, condense=True,
                     precond="multigrid")
    print ("p=", p, ": ndof,nsteps=", r)

 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.328876
1 2.04668e-17
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.356848
1 0.00912278
2 8.39618e-05
3 2.12427e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.35568
1 0.0107157
2 0.000587028
3 2.70143e-05
4 6.53175e-07
5 2.35622e-08
6 5.52189e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35423
1 0.0116601
2 0.000744241
3 3.8359e-05
4 1.65924e-06
5 1.17605e-07
6 6.9037e-09
p= 1 7 4.84505e-10
: ndof,nsteps= [(8, 2), (21, 4), (65, 7), (225, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.361394
1 0.0124871
2 0.000100053
3 3.66458e-07
4 1.90233e-20
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.361122
1 0.0117365
2 0.000512379
3 2.37102e-05
4 8.62628e-07
5 1.77994e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.35605
1 0.0119747
2 0.000709088
3 3.47905e-05
4 2.01234e-06
5 1.26119e-07
6 7.23945e-09
7 2.83652e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35426
1 0.0120479
2 0.000804714
3 5.21269e-05
4 2.59939e-06
5 1.85087e-07
6 1.16487e-08
p= 2 : ndof,nsteps= [(21, 5), (65, 6), (225, 8), (833, 8)]
7 8.25998e-10
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.357462
1 0.0104355
2 3.87107e-05
3 7.03724e-08
4 3.62676e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.360249
1 0.00935938
2 0.000370448
3 8.4552e-06
4 1.74655e-07
5 4.72239e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.356015
1 0.0100516
2 0.000579891
3 3.2525e-05
4 1.71509e-06
5 6.53192e-08
6 2.35908e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.354277
1 0.0113762
2 0.000683641
3 4.38573e-05
4 2.00453e-06
5 1.02697e-07
6 7.00714e-09
p=7 3.88935e-10
 3 : ndof,nsteps= [(40, 5), (133, 6), (481, 7), (1825, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.357282
1 0.0106946
2 4.30855e-05
3 7.85377e-08
4 2.96302e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.36024
1 0.00957924
2 0.000379842
3 9.40357e-06
4 1.94328e-07
5 3.68313e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.355997
1 0.0101793
2 0.000585766
3 3.2499e-05
4 1.75093e-06
5 6.81548e-08
6 2.53159e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35427
1 0.011417
2 0.00069193
3 4.43115e-05
4 2.01881e-06
5 1.04497e-07
6 7.37309e-09
p=7 4.15677e-10
 4 : ndof,nsteps= [(65, 5), (225, 6), (833, 7), (3201, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.35726
1 0.0106856
2 4.24049e-05
3 7.13818e-08
4 1.96349e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.360239
1 0.00955702
2 0.000379561
3 9.27747e-06
4 1.86071e-07
5 3.2234e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.355996
1 0.0101711
2 0.000585526
3 3.25143e-05
4 1.74422e-06
5 6.7538e-08
6 2.4907e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35427
1 0.0114135
2 0.000691626
3 4.42956e-05
4 2.01444e-06
5 1.03847e-07
6 7.26988e-09
p=7 4.10023e-10
 5 : ndof,nsteps= [(96, 5), (341, 6), (1281, 7), (4961, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.357255
1 0.0106908
2 4.23066e-05
3 6.88718e-08
4 1.93619e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.360238
1 0.0095494
2 0.000379264
3 9.23028e-06
4 1.8417e-07
5 3.15399e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.355996
1 0.0101665
2 0.000585296
3 3.25194e-05
4 1.74202e-06
5 6.73754e-08
6 2.48051e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35427
1 0.0114117
2 0.000691289
3 4.42738e-05
4 2.01286e-06
5 1.03681e-07
6 7.24619e-09
p=7 4.0837e-10
 6 : ndof,nsteps= [(133, 5), (481, 6), (1825, 7), (7105, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.357254
1 0.0106922
2 4.2244e-05
3 6.76462e-08
4 1.88383e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.360238
1 0.00954668
2 0.000379188
3 9.21153e-06
4 1.8309e-07
5 3.10225e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.355996
1 0.0101651
2 0.000585253
3 3.25212e-05
4 1.74116e-06
5 6.72989e-08
6 2.47505e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35427
1 0.0114111
2 0.000691218
3 4.42706e-05
4 2.01222e-06
5 1.03591e-07
6 7.23181e-09
7 4.07519e-10
p= 7 : ndof,nsteps= [(176, 5), (645, 6), (2465, 7), (9633, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.357253
1 0.0106929
2 4.22143e-05
3 6.70625e-08
4 1.861e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.360238
1 0.0095452
2 0.000379144
3 9.2017e-06
4 1.82592e-07
5 3.08177e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.355996
1 0.0101644
2 0.000585227
3 3.25221e-05
4 1.74073e-06
5 6.72634e-08
6 2.47259e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35427
1 0.0114108
2 0.000691174
3 4.42684e-05
4 2.01189e-06
5 1.0355e-07
6 7.22548e-09
p=7 4.07128e-10
 8 : ndof,nsteps= [(225, 5), (833, 6), (3201, 7), (12545, 8)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
creating low order biform on demand
assemble VOL element 6/6
assemble VOL element 6/6
call pardiso ... done
BlockJacobi Preconditioner, constructor called, #blocks = 8
BlockJacobi Preconditioner built
assemble VOL element 6/6
0 0.357253
1 0.0106933
2 4.21986e-05
3 6.67556e-08
4 1.84997e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
assemble VOL element 24/24
BlockJacobi Preconditioner, constructor called, #blocks = 21
BlockJacobi Preconditioner built
assemble VOL element 24/24
0 0.360238
1 0.0095443
2 0.000379117
3 9.19615e-06
4 1.82323e-07
5 3.07079e-10
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
assemble VOL element 96/96
BlockJacobi Preconditioner, constructor called, #blocks = 65
BlockJacobi Preconditioner built
assemble VOL element 96/96
0 0.355996
1 0.010164
2 0.00058521
3 3.25226e-05
4 1.74049e-06
5 6.72438e-08
6 2.47123e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
assemble VOL element 384/384
BlockJacobi Preconditioner, constructor called, #blocks = 225
BlockJacobi Preconditioner built
assemble VOL element 384/384
0 0.35427
1 0.0114106
2 0.000691147
3 4.42671e-05
4 2.01171e-06
5 1.03527e-07
6 7.22194e-09
p=7 4.06908e-10
 9 : ndof,nsteps= [(280, 5), (1045, 6), (4033, 7), (15841, 8)]

Element-wise BDDC preconditioner

A built-in element-wise BDDC (Balancing Domain Decomposition preconditioner with Constraints) preconditioner is also available. In contrast to local or multigrid preconditioners, the BDDC preconditioner needs access to the element matrices. This is exactly the reason why we register the preconditioner with the bilinear form bfa before calling bfa.Assemble().

for p in range(1,10):
    r = SolveProblem(h=0.5, p=p, levels=4, condense=True,
                     precond="bddc")
    print ("p=", p, ": ndof,nsteps=", r)

 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 8 )
has inverse
assemble VOL element 6/6
0 0.328876
1 3.04047e-17
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 21 )
has inverse
assemble VOL element 24/24
0 0.364342
1 1.27966e-16
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 65 )
has inverse
assemble VOL element 96/96
0 0.372325
1 3.35155e-16
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 225 )
has inverse
assemble VOL element 384/384
0 0.374289
p=1 1.01645e-15
 1 : ndof,nsteps= [(8, 2), (21, 2), (65, 2), (225, 2)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 21 )
has inverse
assemble VOL element 6/6
0 0.477772
1 0.160558
2 0.0510646
3 0.00871385
4 0.000584462
5 4.49972e-06
6 4.3786e-18
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 65 )
has inverse
assemble VOL element 24/24
0 0.504269
1 0.161055
2 0.0535376
3 0.0155192
4 0.00407181
5 0.000895811
6 0.000138018
7 2.99725e-05
8 2.25761e-06
9 2.97198e-07
10 2.23181e-08
11 2.69378e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 225 )
has inverse
assemble VOL element 96/96
0 0.521382
1 0.138587
2 0.0457176
3 0.0163883
4 0.00512691
5 0.00174602
6 0.000354238
7 0.000102799
8 2.62412e-05
9 8.83994e-06
10 2.76421e-06
11 6.83615e-07
12 1.63929e-07
13 5.20907e-08
14 1.25727e-08
15 3.26535e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 833 )
has inverse
assemble VOL element 384/384
0 0.526756
1 0.129923
2 0.0381012
3 0.0121828
4 0.00482539
5 0.00167383
6 0.000461273
7 0.000111294
8 4.18584e-05
9 1.01023e-05
10 3.5541e-06
11 1.05233e-06
12 3.20884e-07
13 8.72507e-08
14 3.56346e-08
15 8.70265e-09
p=16 3.29933e-09
 2 : ndof,nsteps= [(21, 7), (65, 12), (225, 16), (833, 17)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 40 )
has inverse
assemble VOL element 6/6
0 0.538396
1 0.210873
2 0.0879634
3 0.0267507
4 0.00782763
5 0.000639145
6 0.000135815
7 1.05199e-05
8 2.81451e-07
9 1.83522e-08
10 3.77804e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 133 )
has inverse
assemble VOL element 24/24
0 0.566433
1 0.214309
2 0.117177
3 0.0315078
4 0.00916274
5 0.00431726
6 0.00110362
7 0.000256952
8 6.81979e-05
9 1.71509e-05
10 3.71959e-06
11 1.07889e-06
12 1.67951e-07
13 5.78423e-08
14 8.77879e-09
15 1.71527e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 481 )
has inverse
assemble VOL element 96/96
0 0.586003
1 0.198062
2 0.10448
3 0.0463745
4 0.0127243
5 0.00507468
6 0.00213804
7 0.000896276
8 0.000307158
9 0.000163241
10 5.93704e-05
11 1.73708e-05
12 6.46523e-06
13 2.17002e-06
14 6.99368e-07
15 2.97189e-07
16 1.04997e-07
17 3.54164e-08
18 1.27444e-08
19 4.85557e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 1825 )
has inverse
assemble VOL element 384/384
0 0.592874
1 0.189768
2 0.0949962
3 0.0471908
4 0.0122852
5 0.00449476
6 0.00212718
7 0.0011022
8 0.000357455
9 0.000181719
10 7.63448e-05
11 2.75694e-05
12 9.5549e-06
13 4.45332e-06
14 2.03726e-06
15 6.70775e-07
16 2.39874e-07
17 1.05427e-07
18 4.5521e-08
19 1.67657e-08
20 7.69412e-09
21 3.30288e-09
p= 3 : ndof,nsteps= [(40, 11), (133, 16), (481, 20), (1825, 22)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 65 )
has inverse
assemble VOL element 6/6
0 0.579072
1 0.252102
2 0.134624
3 0.0477871
4 0.0152917
5 0.00332591
6 0.000773557
7 8.34087e-05
8 1.62253e-05
9 8.65157e-07
10 1.3029e-07
11 9.06571e-09
12 6.99862e-11
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 225 )
has inverse
assemble VOL element 24/24
0 0.609904
1 0.256911
2 0.174643
3 0.0536048
4 0.0189014
5 0.00993219
6 0.00340957
7 0.00145839
8 0.000409067
9 0.000167085
10 6.03849e-05
11 1.11345e-05
12 3.21806e-06
13 1.36005e-06
14 2.91481e-07
15 7.07279e-08
16 1.10572e-08
17 1.0365e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 833 )
has inverse
assemble VOL element 96/96
0 0.631966
1 0.243138
2 0.157778
3 0.076626
4 0.024349
5 0.01215
6 0.00643626
7 0.00300594
8 0.0013359
9 0.000761677
10 0.00028142
11 0.000114445
12 5.19383e-05
13 1.80504e-05
14 8.9318e-06
15 4.29427e-06
16 1.57825e-06
17 5.70265e-07
18 2.50238e-07
19 1.05362e-07
20 4.82009e-08
21 2.28613e-08
22 7.59122e-09
23 2.70043e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 3201 )
has inverse
assemble VOL element 384/384
0 0.639866
1 0.236012
2 0.144676
3 0.0779338
4 0.0231727
5 0.0110114
6 0.00680143
7 0.00355654
8 0.00128122
9 0.000775456
10 0.000338054
11 0.000158651
12 8.11772e-05
13 4.2045e-05
14 1.87495e-05
15 8.09238e-06
16 3.57949e-06
17 1.60461e-06
18 6.85347e-07
19 2.8271e-07
20 1.44601e-07
21 6.57553e-08
22 3.0354e-08
23 1.55651e-08
24 7.40174e-09
25 3.05071e-09
p= 4 : ndof,nsteps= [(65, 13), (225, 18), (833, 24), (3201, 26)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 96 )
has inverse
assemble VOL element 6/6
0 0.609638
1 0.27786
2 0.168905
3 0.067266
4 0.0246095
5 0.00743804
6 0.0016936
7 0.000239816
8 5.29463e-05
9 6.84636e-06
10 5.78756e-07
11 1.09465e-07
12 9.43624e-09
13 2.21595e-10
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 341 )
has inverse
assemble VOL element 24/24
0 0.642629
1 0.284212
2 0.219302
3 0.0735349
4 0.0301522
5 0.0165539
6 0.0070891
7 0.00346584
8 0.00100806
9 0.000453504
10 0.000154189
11 4.90608e-05
12 1.40924e-05
13 6.42678e-06
14 1.70074e-06
15 4.35181e-07
16 1.08608e-07
17 2.05711e-08
18 2.84199e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 1281 )
has inverse
assemble VOL element 96/96
0 0.666581
1 0.271872
2 0.200525
3 0.103112
4 0.037077
5 0.0219373
6 0.0128334
7 0.00613042
8 0.00307462
9 0.00173601
10 0.000679267
11 0.000313132
12 0.000135751
13 5.31431e-05
14 3.09029e-05
15 1.49365e-05
16 6.64773e-06
17 3.04055e-06
18 1.47946e-06
19 6.54685e-07
20 3.05687e-07
21 1.30895e-07
22 5.33123e-08
23 2.84076e-08
24 1.09347e-08
25 4.84093e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 4961 )
has inverse
assemble VOL element 384/384
0 0.675292
1 0.26526
2 0.185191
3 0.105087
4 0.0353734
5 0.0210512
6 0.0143058
7 0.007034
8 0.00285275
9 0.00192568
10 0.000844436
11 0.000448509
12 0.000223763
13 0.000108005
14 5.04366e-05
15 2.4986e-05
16 1.17123e-05
17 6.09045e-06
18 3.39771e-06
19 1.80034e-06
20 9.74434e-07
21 4.63004e-07
22 2.09619e-07
23 9.78074e-08
24 5.15415e-08
25 2.84767e-08
26 1.61861e-08
27 7.78844e-09
28 3.71566e-09
p= 5 : ndof,nsteps= [(96, 14), (341, 19), (1281, 26), (4961, 29)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 133 )
has inverse
assemble VOL element 6/6
0 0.634052
1 0.296896
2 0.199359
3 0.0854188
4 0.034219
5 0.0119909
6 0.00264306
7 0.000468708
8 0.000105688
9 2.22183e-05
10 2.29045e-06
11 5.23656e-07
12 8.90097e-08
13 5.74043e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 481 )
has inverse
assemble VOL element 24/24
0 0.668584
1 0.304166
2 0.256316
3 0.0916332
4 0.0412814
5 0.023471
6 0.0114886
7 0.00585759
8 0.00180411
9 0.000860249
10 0.000307832
11 0.000133262
12 3.52689e-05
13 1.82801e-05
14 5.58085e-06
15 1.77709e-06
16 4.47392e-07
17 1.17169e-07
18 1.98227e-08
19 4.57706e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 1825 )
has inverse
assemble VOL element 96/96
0 0.693946
1 0.293149
2 0.236418
3 0.126704
4 0.0492331
5 0.0321474
6 0.0197148
7 0.00958778
8 0.00511968
9 0.00288448
10 0.00118241
11 0.000581591
12 0.000258099
13 0.000118729
14 7.70116e-05
15 3.8441e-05
16 1.93609e-05
17 8.91655e-06
18 4.4225e-06
19 2.14651e-06
20 1.11278e-06
21 5.42801e-07
22 2.82295e-07
23 1.28917e-07
24 5.19936e-08
25 2.57514e-08
26 1.14726e-08
27 5.96676e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 7105 )
has inverse
assemble VOL element 384/384
0 0.70326
1 0.287074
2 0.219414
3 0.129194
4 0.047018
5 0.031483
6 0.0220712
7 0.0106544
8 0.00467279
9 0.00327821
10 0.00144992
11 0.000808523
12 0.000403251
13 0.000200197
14 0.000104347
15 6.02458e-05
16 3.10128e-05
17 1.86832e-05
18 1.11331e-05
19 5.67534e-06
20 2.92962e-06
21 1.47091e-06
22 7.32986e-07
23 4.04825e-07
24 2.62185e-07
25 1.5301e-07
26 7.86562e-08
27 3.84926e-08
28 2.1601e-08
29 1.04566e-08
30 5.79499e-09
p= 6 : ndof,nsteps= [(133, 14), (481, 20), (1825, 28), (7105, 31)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 176 )
has inverse
assemble VOL element 6/6
0 0.654148
1 0.311678
2 0.224394
3 0.101608
4 0.0436865
5 0.0166895
6 0.00364509
7 0.000763652
8 0.000183562
9 5.09557e-05
10 6.45777e-06
11 1.27704e-06
12 3.00989e-07
13 2.74307e-08
14 1.59531e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 645 )
has inverse
assemble VOL element 24/24
0 0.690033
1 0.319916
2 0.287228
3 0.108091
4 0.052324
5 0.0304177
6 0.0162335
7 0.00850322
8 0.00274402
9 0.00138897
10 0.000528585
11 0.000276074
12 6.98573e-05
13 4.10207e-05
14 1.35361e-05
15 4.94268e-06
16 1.17082e-06
17 3.72784e-07
18 6.83441e-08
19 1.60658e-08
20 1.25112e-08
21 3.28152e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 2465 )
has inverse
assemble VOL element 96/96
0 0.716568
1 0.309958
2 0.266749
3 0.147727
4 0.0608063
5 0.0423413
6 0.0266953
7 0.0132158
8 0.00737877
9 0.00416582
10 0.00179288
11 0.00093555
12 0.00043922
13 0.000234402
14 0.000159743
15 8.04139e-05
16 4.29773e-05
17 1.99506e-05
18 1.06538e-05
19 5.72396e-06
20 3.1506e-06
21 1.57305e-06
22 8.20582e-07
23 3.3467e-07
24 1.54264e-07
25 8.05309e-08
26 3.78695e-08
27 2.07063e-08
28 8.59129e-09
29 4.25006e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 9633 )
has inverse
assemble VOL element 384/384
0 0.726381
1 0.304301
2 0.248479
3 0.150687
4 0.0580427
5 0.0418748
6 0.0298147
7 0.0143325
8 0.00666651
9 0.00478602
10 0.00215189
11 0.0012531
12 0.00063945
13 0.000341147
14 0.000203596
15 0.000134084
16 7.08705e-05
17 4.4838e-05
18 2.58681e-05
19 1.29912e-05
20 6.98218e-06
21 3.96493e-06
22 2.18234e-06
23 1.30064e-06
24 8.42532e-07
25 4.44741e-07
26 2.18797e-07
27 1.1486e-07
28 6.87551e-08
29 3.33178e-08
30 1.8979e-08
31 1.00739e-08
32 5.55881e-09
p= 7 : ndof,nsteps= [(176, 15), (645, 22), (2465, 30), (9633, 33)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 225 )
has inverse
assemble VOL element 6/6
0 0.671212
1 0.323679
2 0.246663
3 0.116403
4 0.0527519
5 0.0213917
6 0.00468989
7 0.00113197
8 0.000290723
9 9.18845e-05
10 1.4177e-05
11 2.54348e-06
12 6.80372e-07
13 8.58149e-08
14 8.6667e-09
15 6.84491e-10
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 833 )
has inverse
assemble VOL element 24/24
0 0.708262
1 0.332689
2 0.313769
3 0.12293
4 0.0626905
5 0.0372148
6 0.0211258
7 0.0113082
8 0.00379648
9 0.00201614
10 0.000811927
11 0.000471897
12 0.000118854
13 7.69036e-05
14 2.6512e-05
15 1.01165e-05
16 2.38574e-06
17 8.29915e-07
18 1.74298e-07
19 4.79492e-08
20 5.51062e-08
21 9.97264e-09
22 3.30053e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 3201 )
has inverse
assemble VOL element 96/96
0 0.735791
1 0.323656
2 0.292995
3 0.166602
4 0.0715318
5 0.0521726
6 0.0336087
7 0.0169289
8 0.00978968
9 0.0055656
10 0.00250224
11 0.0013784
12 0.000688073
13 0.000407984
14 0.000279994
15 0.000141021
16 7.78284e-05
17 3.69679e-05
18 2.10052e-05
19 1.17865e-05
20 6.46851e-06
21 3.19185e-06
22 1.66343e-06
23 6.81475e-07
24 3.53463e-07
25 1.90953e-07
26 9.47942e-08
27 5.35006e-08
28 2.25515e-08
29 1.30037e-08
30 6.81226e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 12545 )
has inverse
assemble VOL element 384/384
0 0.746024
1 0.318385
2 0.273714
3 0.169974
4 0.0683071
5 0.0519138
6 0.0374135
7 0.018041
8 0.00879457
9 0.0064322
10 0.00295103
11 0.00179056
12 0.000949598
13 0.000552696
14 0.000368821
15 0.000256221
16 0.000131039
17 8.3211e-05
18 4.68846e-05
19 2.40954e-05
20 1.36386e-05
21 8.33145e-06
22 4.6401e-06
23 2.77138e-06
24 1.74989e-06
25 8.98816e-07
26 4.54557e-07
27 2.57715e-07
28 1.60717e-07
29 7.99409e-08
30 4.72749e-08
31 2.59657e-08
32 1.55619e-08
33 9.28695e-09
p= 34 4.77369e-09
8 : ndof,nsteps= [(225, 16), (833, 23), (3201, 31), (12545, 35)]
 Generate Mesh from spline geometry
 Boundary mesh done, np = 8
 CalcLocalH: 8 Points 0 Elements 0 Surface Elements
 Meshing domain 1 / 1
 Surface meshing done
 Edgeswapping, topological
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Edgeswapping, metric
 Smoothing
 Split improve
 Combine improve
 Smoothing
 Update mesh topology
 Update clusters
assemble VOL element 6/6
call wirebasket inverse ( with 3 free dofs out of 280 )
has inverse
assemble VOL element 6/6
0 0.686009
1 0.333687
2 0.265894
3 0.129732
4 0.0614166
5 0.026037
6 0.0057853
7 0.00155793
8 0.00042913
9 0.000142234
10 2.53681e-05
11 4.61896e-06
12 1.20641e-06
13 2.15219e-07
14 2.63988e-08
15 1.50668e-09
 Mesh bisection
 resetting marked-element information
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 24/24
call wirebasket inverse ( with 12 free dofs out of 1045 )
has inverse
assemble VOL element 24/24
0 0.724075
1 0.343461
2 0.336763
3 0.136439
4 0.0726134
5 0.0438518
6 0.0260637
7 0.0142074
8 0.00492846
9 0.00273629
10 0.00115534
11 0.000720277
12 0.000182983
13 0.000126907
14 4.45018e-05
15 1.72924e-05
16 4.24845e-06
17 1.535e-06
18 3.86066e-07
19 1.05082e-07
20 9.9182e-08
21 2.48525e-08
22 8.32822e-09
23 2.35425e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 96/96
call wirebasket inverse ( with 48 free dofs out of 4033 )
has inverse
assemble VOL element 96/96
0 0.752462
1 0.335216
2 0.315934
3 0.183614
4 0.0815573
5 0.0616582
6 0.0403953
7 0.0206709
8 0.0122958
9 0.00704367
10 0.00330209
11 0.00190959
12 0.00100481
13 0.000636805
14 0.000431337
15 0.000217736
16 0.000123077
17 6.00684e-05
18 3.55894e-05
19 2.0301e-05
20 1.10314e-05
21 5.44034e-06
22 2.87833e-06
23 1.23577e-06
24 7.08725e-07
25 3.93271e-07
26 2.04409e-07
27 1.1803e-07
28 5.15348e-08
29 3.25377e-08
30 1.62239e-08
31 7.84744e-09
32 6.00621e-09
 Mesh bisection
 Update mesh topology
 Update clusters
 Bisection done
 Update clusters
assemble VOL element 384/384
call wirebasket inverse ( with 192 free dofs out of 15841 )
has inverse
assemble VOL element 384/384
0 0.763057
1 0.330268
2 0.295856
3 0.187372
4 0.0778783
5 0.0615829
6 0.0448236
7 0.0217297
8 0.0110093
9 0.00817423
10 0.0038355
11 0.00242237
12 0.0013387
13 0.000839546
14 0.000597926
15 0.000416279
16 0.000205558
17 0.000130996
18 7.3347e-05
19 3.89638e-05
20 2.29853e-05
21 1.4629e-05
22 8.07927e-06
23 4.83288e-06
24 3.01951e-06
25 1.56002e-06
26 8.28947e-07
27 5.06704e-07
28 3.27138e-07
29 1.68883e-07
30 1.03369e-07
31 5.91226e-08
32 3.74135e-08
33 2.23794e-08
34 1.18499e-08
p= 9 : ndof,nsteps= [(280, 16), (1045, 24), (4033, 33), (15841, 36)]
35 6.64736e-09

The BDDC preconditioner needs more iterations, but the work per iteration is less, so performance is similar to multigrid. This element-wise BDDC preconditioner works well in shared memory parallel as well as in distributed memory mode. See \(\S\)2.1.3 for more about BDDC preconditioner and how to combine it with an algebraic multigrid coarse solver.