This page was generated from jupyter-files/unit-1.1-poisson/poisson.ipynb.
1.1 First NGSolve example¶
Let us solve the Poisson problem.
\[\text{find: } u \in H_{0,D}^1 \quad \int_\Omega \nabla u \nabla v = \int_\Omega f v, \quad \text{ for all } v \in H_{0,D}^1.\]
Quick steps to solution:¶
1. Import NGSolve and Netgen Python modules:¶
[1]:
import netgen.gui
%gui tk
from ngsolve import *
from netgen.geom2d import unit_square
2. Generate an unstructured mesh¶
[2]:
mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))
mesh.nv, mesh.ne # number of vertices & elements
[2]:
(39, 56)
- Here we prescribed a maximal mesh-size of 0.2 using the
maxhflag. - The mesh can be viewed by switching to the
Meshtab in the Netgen GUI.
3. Declare a finite element space:¶
[3]:
fes = H1(mesh, order=2, dirichlet="bottom|right")
fes.ndof # number of unknowns in this space
[3]:
133
Python’s help system displays further documentation.
[4]:
help(fes)
Help on H1 in module ngsolve.comp object:
class H1(FESpace)
| Keyword arguments can be:
| order: int = 1
| order of finite element space
| complex: bool = False
| dirichlet: regexpr
| Regular expression string defining the dirichlet boundary.
| More than one boundary can be combined by the | operator,
| i.e.: dirichlet = 'top|right'
| definedon: Region or regexpr
| FESpace is only defined on specific Region, created with mesh.Materials('regexpr')
| or mesh.Boundaries('regexpr'). If given a regexpr, the region is assumed to be
| mesh.Materials('regexpr').
| dim: int = 1
| Create multi dimensional FESpace (i.e. [H1]^3)
| dgjumps: bool = False
| Enable discontinuous space for DG methods, this flag is needed for DG methods,
| since the dofs have a different coupling then and this changes the sparsity
| pattern of matrices.
|
| Method resolution order:
| H1
| FESpace
| pybind11_builtins.pybind11_object
| builtins.object
|
| Methods defined here:
|
| __getstate__(...)
| __getstate__(self: ngsolve.comp.FESpace) -> tuple
|
| __init__(...)
| __init__(self: ngsolve.comp.H1, mesh: ngsolve.comp.Mesh, **kwargs) -> None
|
| __setstate__(...)
| __setstate__(self: ngsolve.comp.H1, arg0: tuple) -> None
|
| ----------------------------------------------------------------------
| Data descriptors defined here:
|
| __dict__
|
| ----------------------------------------------------------------------
| Methods inherited from FESpace:
|
| ApplyM(...)
| ApplyM(self: ngsolve.comp.FESpace, vec: ngsolve.la.BaseVector, rho: ngsolve.fem.CoefficientFunction=None) -> None
|
| Apply mass-matrix. Available only for L2-like spaces
|
| CouplingType(...)
| CouplingType(self: ngsolve.comp.FESpace, dofnr: int) -> ngsolve.comp.COUPLING_TYPE
|
| get coupling type of a degree of freedom
|
| Elements(...)
| Elements(self: ngsolve.comp.FESpace, VOL_or_BND: ngsolve.comp.VorB=VorB.VOL) -> ngsolve.comp.FESpaceElementRange
|
| FinalizeUpdate(...)
| FinalizeUpdate(self: ngsolve.comp.FESpace) -> None
|
| finalize update
|
| FreeDofs(...)
| FreeDofs(self: ngsolve.comp.FESpace, coupling: bool=False) -> ngsolve.ngstd.BitArray
|
| Return BitArray of free (non-Dirichlet) dofs
| coupling=False ... all free dofs including local dofs
| coupling=True .... only element-boundary free dofs
|
| GetDofNrs(...)
| GetDofNrs(*args, **kwargs)
| Overloaded function.
|
| 1. GetDofNrs(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.ElementId) -> tuple
|
| 2. GetDofNrs(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.NodeId) -> tuple
|
| GetDofs(...)
| GetDofs(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.Region) -> ngsolve.ngstd.BitArray
|
| GetFE(...)
| GetFE(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.ElementId) -> object
|
| ParallelDofs(...)
| ParallelDofs(self: ngsolve.comp.FESpace) -> ngsolve.la.ParallelDofs
|
| Return dof-identification for MPI-distributed meshes
|
| Range(...)
| Range(self: ngsolve.comp.FESpace, component: int) -> slice
|
| Return interval of dofs of a component of a product space
|
| SetCouplingType(...)
| SetCouplingType(self: ngsolve.comp.FESpace, dofnr: int, coupling_type: ngsolve.comp.COUPLING_TYPE) -> None
|
| set coupling type of a degree of freedom
|
| SetDefinedOn(...)
| SetDefinedOn(self: ngsolve.comp.FESpace, Region: ngsolve.comp.Region) -> None
|
| SetOrder(...)
| SetOrder(*args, **kwargs)
| Overloaded function.
|
| 1. SetOrder(self: ngsolve.comp.FESpace, element_type: ngsolve.fem.ET, order: int) -> None
|
| 2. SetOrder(self: ngsolve.comp.FESpace, nodeid: ngsolve.comp.NodeId, order: int) -> None
|
| SolveM(...)
| SolveM(self: ngsolve.comp.FESpace, vec: ngsolve.la.BaseVector, rho: ngsolve.fem.CoefficientFunction=None) -> None
|
| Solve with the mass-matrix. Available only for L2-like spaces
|
| TestFunction(...)
| TestFunction(self: ngsolve.comp.FESpace) -> object
|
| Return a proxy to be used as a testfunction for Symbolic Integrators
|
| TnT(...)
| TnT(self: ngsolve.comp.FESpace) -> Tuple[object, object]
|
| Return a tuple of trial and testfunction
|
| TrialFunction(...)
| TrialFunction(self: ngsolve.comp.FESpace) -> object
|
| Return a proxy to be used as a trialfunction in Symbolic Integrators
|
| Update(...)
| Update(self: ngsolve.comp.FESpace) -> None
|
| update space after mesh-refinement
|
| UpdateDofTables(...)
| UpdateDofTables(self: ngsolve.comp.FESpace) -> None
|
| update dof-tables after changing polynomial order distribution
|
| __eq__(...)
| __eq__(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace) -> bool
|
| __flags_doc__(...) from builtins.PyCapsule
| __flags_doc__() -> dict
|
| __special_treated_flags__(...) from builtins.PyCapsule
| __special_treated_flags__() -> dict
|
| __str__(...)
| __str__(self: ngsolve.comp.FESpace) -> str
|
| __timing__(...)
| __timing__(self: ngsolve.comp.FESpace) -> object
|
| ----------------------------------------------------------------------
| Data descriptors inherited from FESpace:
|
| components
| Return a list of the components of a product space
|
| globalorder
| query global order of space
|
| is_complex
|
| lospace
|
| mesh
|
| ndof
| number of degrees of freedom
|
| ndofglobal
| global number of dofs on MPI-distributed mesh
|
| type
| type of finite element space
|
| ----------------------------------------------------------------------
| Methods inherited from pybind11_builtins.pybind11_object:
|
| __new__(*args, **kwargs) from pybind11_builtins.pybind11_type
| Create and return a new object. See help(type) for accurate signature.
4. Declare test function, trial function, and grid function¶
- Test and trial function are symbolic objects - called
ProxyFunctions- that help you construct bilinear forms (and have no space to hold solutions). GridFunctions, on the other hand, represent functions in the finite element space and contains memory to hold coefficient vectors.
[5]:
u = fes.TrialFunction() # symbolic object
v = fes.TestFunction() # symbolic object
gfu = GridFunction(fes) # solution
A shorter command for obtaining at once both trial and test variables is now available:
[6]:
u, v = fes.TnT()
5. Define and assemble linear and bilinear forms:¶
[7]:
a = BilinearForm(fes, symmetric=True)
a += SymbolicBFI(grad(u)*grad(v))
a.Assemble()
f = LinearForm(fes)
f += SymbolicLFI(x*v)
f.Assemble()
You can examine the linear system in more detail:
[8]:
print(f.vec)
0.000333333
0.0090505
0.00633333
0.000770891
0.00419003
0.00873851
0.0110113
0.0118831
0.0154585
0.0172657
0.0153513
0.0180248
0.0203006
0.00930857
0.00722848
0.00380471
0.000687974
0.000703175
0.00144642
0.00174641
0.0130156
0.0196218
0.0235625
0.0209991
0.032979
0.0275741
0.0284488
0.0170589
0.0176338
0.0124153
0.0051304
0.00284308
0.00630915
0.0220354
0.0157804
0.0210096
0.0100578
0.0181327
0.0217548
-6.66667e-05
-3.33333e-05
-0.000536217
-0.000585765
-0.00111116
-0.0008
-0.000766667
-8.25863e-05
-2.35728e-05
-0.000125108
-0.000290565
-0.000209771
-0.000451004
-0.00040564
-0.000661732
-0.000800436
-0.000504522
-0.000864384
-0.000967242
-0.000947681
-0.000970631
-0.000782608
-0.00113705
-0.00131313
-0.000661412
-0.00146165
-0.00135486
-0.000642205
-0.00125381
-0.0012366
-0.00157121
-0.00143302
-0.000438352
-0.00133842
-0.000991598
-0.000377208
-0.000778967
-0.000731847
-0.000237359
-0.000666891
-0.000536209
-0.00039775
-0.000251178
-1.37451e-05
-9.64747e-05
-7.25997e-05
-2.40468e-05
-7.40193e-05
-9.91414e-05
-4.37045e-05
-0.000208445
-0.000157729
-0.000237114
-0.000737581
-0.000370323
-0.000591075
-0.00090079
-0.000832185
-0.000759186
-0.000954243
-0.00110341
-0.00101272
-0.00107714
-0.00121445
-0.00106391
-0.00100948
-0.00115025
-0.000971716
-0.000950052
-0.000983723
-0.000968857
-0.000711981
-0.000799923
-0.000661911
-0.000816822
-0.00081628
-0.000348442
-0.000484742
-0.000677524
-0.00016251
-0.000294359
-0.000169603
-0.000232291
-0.000459975
-0.000328541
-0.000828049
-0.000903769
-0.000868363
-0.000543998
-0.000763656
-0.000844269
-0.000848631
-0.000633923
-0.0008243
[9]:
print(a.mat)
Row 0: 0: 1
Row 1: 1: 0.829127
Row 2: 2: 1
Row 3: 3: 0.880753
Row 4: 0: -0.5 4: 1.90755
Row 5: 4: -0.559177 5: 1.91948
Row 6: 5: -0.37742 6: 1.7799
Row 7: 1: -0.222678 6: -0.298266 7: 1.83885
Row 8: 1: -0.223343 8: 1.90243
Row 9: 8: -0.375377 9: 1.75223
Row 10: 9: -0.283322 10: 1.78161
Row 11: 2: -0.5 10: -0.268388 11: 1.99954
Row 12: 2: -0.5 11: -0.098129 12: 1.84322
Row 13: 12: -0.211616 13: 1.75354
Row 14: 13: -0.319115 14: 1.77273
Row 15: 3: -0.359363 14: -0.356559 15: 1.85161
Row 16: 3: -0.354152 16: 1.80192
Row 17: 16: -0.146234 17: 1.85831
Row 18: 17: -0.280268 18: 1.8809
Row 19: 0: -0.5 4: -0.159147 18: -0.497592 19: 1.89108
Row 20: 4: -0.689223 5: -0.134026 18: -0.213492 19: -0.734339 20: 3.5107
Row 21: 5: -0.848855 6: -0.338846 20: -0.607596 21: 3.57058
Row 22: 6: -0.76537 7: -0.325948 21: -0.590849 22: 3.53156
Row 23: 1: -0.383106 7: -0.991955 8: -1.08169 22: -0.711434 23: 3.78819
Row 24: 8: -0.222018 9: -0.654805 22: -0.588928 23: -0.62 24: 3.44463
Row 25: 9: -0.438728 10: -0.852694 24: -0.516043 25: 3.57605
Row 26: 10: -0.377206 11: -1.13302 12: -0.22963 25: -0.541968 26: 3.73524
Row 27: 12: -0.803846 13: -0.666276 26: -0.773272 27: 3.68753
Row 28: 13: -0.556538 14: -0.353124 27: -0.781766 28: 3.60218
Row 29: 14: -0.74393 15: -0.202714 28: -0.943134 29: 3.72717
Row 30: 3: -0.167237 15: -0.932971 16: -0.826096 29: -0.840918 30: 3.7686
Row 31: 16: -0.475439 17: -1.04297 30: -0.848015 31: 3.7525
Row 32: 17: -0.388835 18: -0.889547 20: -0.459182 31: -0.532046 32: 3.57532
Row 33: 21: -0.763143 22: -0.54903 24: -0.350551 33: 3.66745
Row 34: 20: -0.672842 21: -0.421292 32: -0.676988 33: -0.704385 34: 3.51434
Row 35: 25: -0.347729 26: -0.680143 27: -0.662371 28: -0.447405 35: 3.54016
Row 36: 29: -0.830595 30: -0.153364 31: -0.854028 32: -0.628726 34: -0.40922 36: 3.64742
Row 37: 24: -0.492286 25: -0.878889 33: -1.07273 35: -0.76226 37: 3.75396
Row 38: 28: -0.520216 29: -0.165884 33: -0.227614 34: -0.629613 35: -0.640254 36: -0.771484 37: -0.547794 38: 3.50286
Row 39: 0: -0.0833333 4: 0 19: 0.0833333 39: 0.0416667
Row 40: 0: -0.0833333 4: 0.0833333 19: 0 39: 1.73472e-18 40: 0.0416667
Row 41: 1: -0.0318638 7: -0.0835338 23: 0.115398 41: 0.0381276
Row 42: 1: -0.0319873 8: -0.0831334 23: 0.115121 42: 0.0380861
Row 43: 1: -0.0743368 7: 0.120647 8: 0.120357 23: -0.166667 41: -0.0208834 42: -0.0207833 43: 0.0762138
Row 44: 2: -0.0833333 11: 0 12: 0.0833333 44: 0.0416667
Row 45: 2: -0.0833333 11: 0.0833333 12: 0 44: 0 45: 0.0416667
Row 46: 3: -0.0151987 15: -0.0803559 30: 0.0955546 46: 0.0388621
Row 47: 3: -0.0126741 16: -0.0864211 30: 0.0990952 47: 0.0395302
Row 48: 3: -0.118919 15: 0.14025 16: 0.145447 30: -0.166777 46: -0.020089 47: -0.0216053 48: 0.0783923
Row 49: 4: -0.0583984 5: -0.00990762 20: 0.068306 49: 0.0403755
Row 50: 0: 0.166667 4: -0.139806 19: -0.148957 20: 0.122096 39: -0.0208333 40: -0.0208333 50: 0.0788218
Row 51: 4: -0.119721 5: 0.103104 19: 0.0921482 20: -0.0755312 49: -0.0024769 50: -0.0164059 51: 0.0775306
Row 52: 5: -0.0550429 6: -0.0295225 21: 0.0845653 52: 0.0368672
Row 53: 4: 0.151595 5: -0.179629 20: -0.117774 21: 0.145809 49: -0.0145996 51: -0.023299 53: 0.0799353
Row 54: 5: -0.0753334 6: 0.0924258 20: 0.0718059 21: -0.0888984 52: -0.00738062 53: -0.014844 54: 0.0764269
Row 55: 6: -0.0582582 7: -0.0374957 22: 0.0957539 55: 0.0363662
Row 56: 5: 0.117946 6: -0.132207 21: -0.107784 22: 0.122044 52: -0.0137607 54: -0.0157258 56: 0.0741162
Row 57: 6: -0.0766627 7: 0.0872066 21: 0.0796928 22: -0.0902367 55: -0.00937392 56: -0.0131853 57: 0.0736152
Row 58: 6: 0.107969 7: -0.131503 22: -0.114716 23: 0.13825 55: -0.0145646 57: -0.0124277 58: 0.075136
Row 59: 1: 0.0689767 7: -0.053942 22: 0.0732872 23: -0.0883219 41: -0.00796594 43: -0.00927824 58: -0.0141145 59: 0.0768974
Row 60: 8: -0.0282254 9: -0.0570403 24: 0.0852656 60: 0.0369571
Row 61: 1: 0.0692111 8: -0.0460015 23: -0.0894962 24: 0.0662866 42: -0.00799682 43: -0.00930597 61: 0.0789449
Row 62: 8: -0.159711 9: 0.119603 23: 0.154658 24: -0.114549 60: -0.0142601 61: -0.0143772 62: 0.0778159
Row 63: 9: -0.0300697 10: -0.071478 25: 0.101548 63: 0.037192
Row 64: 8: 0.0907882 9: -0.105614 24: -0.0776415 25: 0.0924677 60: -0.00705634 62: -0.0156407 64: 0.0730975
Row 65: 9: -0.0993143 10: 0.118698 24: 0.10151 25: -0.120894 63: -0.0178695 64: -0.012354 65: 0.0733324
Row 66: 10: -0.0304682 11: -0.0742233 26: 0.104691 66: 0.0373557
Row 67: 9: 0.0772901 10: -0.0796198 25: -0.0752556 26: 0.0775854 63: -0.00751743 65: -0.0118051 67: 0.0742478
Row 68: 10: -0.115369 11: 0.118955 25: 0.115823 26: -0.119409 66: -0.0185558 67: -0.0112965 68: 0.0744115
Row 69: 2: 0.166667 11: -0.197946 12: -0.122045 26: 0.153325 44: -0.0208333 45: -0.0208333 69: 0.0840865
Row 70: 10: 0.0751996 11: -0.0610862 12: 0.0550663 26: -0.0691797 66: -0.00761705 68: -0.0111829 69: -0.00967788 70: 0.0797756
Row 71: 12: -0.0509965 13: -0.0596508 27: 0.110647 71: 0.0364792
Row 72: 11: 0.130968 12: -0.0993327 26: -0.199191 27: 0.167556 69: -0.0286533 70: -0.00408871 72: 0.084199
Row 73: 12: -0.0348296 13: 0.0949202 26: 0.0841386 27: -0.144229 71: -0.0149127 72: -0.0211446 73: 0.0782583
Row 74: 13: -0.0597076 14: -0.0333842 28: 0.0930917 74: 0.0365694
Row 75: 12: 0.0862659 13: -0.0683181 27: -0.113119 28: 0.0951715 71: -0.0127491 73: -0.00881734 75: 0.0731209
Row 76: 13: -0.104581 14: 0.08657 27: 0.113518 28: -0.095507 74: -0.00834605 75: -0.0155307 76: 0.0732111
Row 77: 14: -0.0676137 15: -0.0230352 29: 0.0906489 77: 0.0375189
Row 78: 13: 0.112893 14: -0.10956 28: -0.127013 29: 0.12368 74: -0.0149269 76: -0.0132964 78: 0.0738569
Row 79: 14: -0.0848962 15: 0.0824617 28: 0.0927755 29: -0.090341 77: -0.00575881 78: -0.0168264 79: 0.0748064
Row 80: 14: 0.12704 15: -0.134566 29: -0.139062 30: 0.146588 77: -0.0169034 79: -0.0148566 80: 0.0768533
Row 81: 3: 0.0750926 15: -0.0706443 29: 0.0821986 30: -0.086647 46: -0.00379968 48: -0.0149735 80: -0.0178621 81: 0.0781966
Row 82: 16: -0.0441816 17: -0.0855914 31: 0.129773 82: 0.0385364
Row 83: 3: 0.0716995 16: -0.0940836 30: -0.0723049 31: 0.0946889 47: -0.00316853 48: -0.0147563 83: 0.0760178
Row 84: 16: -0.0756339 17: 0.109964 30: 0.110892 31: -0.145222 82: -0.0213979 83: -0.0149077 84: 0.075024
Row 85: 17: -0.042337 18: -0.0557767 32: 0.0981137 85: 0.0362063
Row 86: 16: 0.068554 17: -0.0468412 31: -0.0890017 32: 0.0672889 82: -0.0110454 84: -0.00609309 86: 0.0774179
Row 87: 17: -0.134949 18: 0.102488 31: 0.133057 32: -0.100597 85: -0.0139442 86: -0.011205 87: 0.0750878
Row 88: 18: -0.016011 19: -0.0567662 20: 0.0727772 88: 0.0389273
Row 89: 18: -0.175413 19: 0.139698 20: -0.102588 32: 0.138303 88: -0.0141915 89: 0.0783959
Row 90: 17: 0.0890484 18: -0.0662824 20: 0.0653934 32: -0.0881593 85: -0.0105842 87: -0.0116778 89: -0.0114556 90: 0.0756749
Row 91: 4: 0.0829967 18: 0.0989431 19: -0.109457 20: -0.0724832 50: -0.014118 51: -0.00663113 88: -0.00400276 89: -0.020733 91: 0.0760824
Row 92: 5: 0.098863 20: -0.0714262 21: -0.130771 34: 0.103334 53: -0.0216082 54: -0.0031075 92: 0.0758658
Row 93: 18: 0.112052 20: -0.0727152 32: -0.155837 34: 0.1165 89: -0.0231203 90: -0.00489276 93: 0.0762705
Row 94: 20: -0.072598 21: 0.086228 32: 0.0940636 34: -0.107694 92: -0.0110845 93: -0.0158389 94: 0.073108
Row 95: 6: 0.0962552 21: -0.0854376 22: -0.110271 33: 0.0994538 56: -0.0173258 57: -0.00673795 95: 0.0735459
Row 96: 21: -0.0716112 22: 0.0867017 33: -0.095593 34: 0.0805024 95: -0.010242 96: 0.0735987
Row 97: 20: 0.100886 21: -0.110595 33: 0.12333 34: -0.113621 92: -0.014749 94: -0.0104725 96: -0.0136563 97: 0.0736078
Row 98: 7: 0.0986211 22: -0.0547959 23: -0.127616 24: 0.0837913 58: -0.020448 59: -0.00420727 98: 0.0752462
Row 99: 22: -0.112651 23: 0.107939 24: -0.0810715 33: 0.0857841 98: -0.0114561 99: 0.0729694
Row 100: 21: 0.10422 22: -0.105922 24: 0.0954349 33: -0.093733 95: -0.0146215 96: -0.0114335 99: -0.00881178 100: 0.0727899
Row 101: 8: 0.105926 22: 0.100081 23: -0.159263 24: -0.0467445 61: -0.00219441 62: -0.0242872 98: -0.00949172 99: -0.0155286 101: 0.0773352
Row 102: 9: 0.0951455 24: -0.0879729 25: -0.120761 37: 0.113588 64: -0.0107629 65: -0.0130235 102: 0.0736852
Row 103: 22: 0.110725 24: -0.106357 33: -0.135247 37: 0.130879 99: -0.0126343 100: -0.015047 103: 0.0750071
Row 104: 24: -0.059769 25: 0.1143 33: 0.107888 37: -0.162419 102: -0.0194273 103: -0.0211775 104: 0.076059
Row 105: 10: 0.103037 25: -0.106824 26: -0.097702 35: 0.101489 67: -0.00809987 68: -0.0176594 105: 0.0737134
Row 106: 25: -0.113914 26: 0.110445 35: -0.125467 37: 0.128937 105: -0.0163256 106: 0.074334
Row 107: 24: 0.07247 25: -0.0583599 35: 0.0819333 37: -0.0960434 102: -0.00896975 104: -0.00914776 106: -0.0150411 107: 0.0752212
Row 108: 12: 0.0825381 26: -0.0476148 27: -0.13512 35: 0.100197 72: -0.0207445 73: 0.000109951 108: 0.0779035
Row 109: 25: 0.0813281 26: -0.0894424 27: 0.0964427 35: -0.0883284 105: -0.0090465 106: -0.0112855 108: -0.0130356 109: 0.072782
Row 110: 13: 0.084444 27: -0.109648 28: -0.056567 35: 0.0817709 75: -0.00826218 76: -0.0128488 110: 0.0741273
Row 111: 26: 0.0923549 27: -0.112472 28: 0.0916897 35: -0.0715729 108: -0.0120137 109: -0.0110751 110: -0.00587956 111: 0.0736099
Row 112: 14: 0.0818443 28: -0.0773065 29: -0.0724993 38: 0.0679614 78: -0.0140936 79: -0.00636744 112: 0.0767487
Row 113: 27: 0.126424 28: -0.103037 35: -0.118365 38: 0.0949784 110: -0.0145632 111: -0.0170429 113: 0.0739925
Row 114: 28: -0.140932 29: 0.106008 35: 0.111162 38: -0.0762371 112: -0.00403117 113: -0.0150281 114: 0.0759681
Row 115: 15: 0.0858897 29: -0.0709417 30: -0.0969322 36: 0.0819842 80: -0.0187848 81: -0.00268759 115: 0.0770067
Row 116: 29: -0.0802273 30: 0.0904976 36: -0.0891129 38: 0.0788427 115: -0.00544822 116: 0.0769432
Row 117: 28: 0.14172 29: -0.168124 36: 0.145561 38: -0.119157 112: -0.0129592 114: -0.0224708 116: -0.01683 117: 0.0787321
Row 118: 16: 0.0863197 30: -0.0550293 31: -0.112704 36: 0.0814134 83: -0.00876454 84: -0.0128154 118: 0.0772672
Row 119: 29: 0.128896 30: -0.15041 31: 0.159351 36: -0.137837 115: -0.0150478 116: -0.0171762 118: -0.0194114 119: 0.0784519
Row 120: 17: 0.110706 31: -0.15293 32: -0.0603088 36: 0.102532 86: -0.0056172 87: -0.0220593 120: 0.0754782
Row 121: 30: 0.085473 31: -0.125559 32: 0.0816942 36: -0.0416078 118: -0.000941958 119: -0.0204263 120: -0.00945999 121: 0.0773763
Row 122: 20: 0.083852 32: -0.0976556 34: -0.084342 36: 0.0981455 93: -0.013286 94: -0.00767698 122: 0.0737072
Row 123: 31: 0.108547 32: -0.09333 34: 0.0806736 36: -0.0958903 120: -0.0161731 121: -0.0109636 122: -0.00779947 123: 0.073502
Row 124: 21: 0.094582 33: -0.0965577 34: -0.0832128 38: 0.0851884 96: -0.00646935 97: -0.0171762 124: 0.074292
Row 125: 24: 0.0693467 33: -0.0332607 37: -0.103733 38: 0.0676465 103: -0.0115422 104: -0.0057945 125: 0.0789452
Row 126: 33: -0.15685 34: 0.120108 37: 0.151642 38: -0.114899 124: -0.0143338 125: -0.014391 126: 0.0774213
Row 127: 32: 0.116423 34: -0.0970758 36: -0.128208 38: 0.108861 122: -0.0167369 123: -0.0123689 127: 0.0733718
Row 128: 33: 0.0906254 34: -0.0997778 36: 0.098266 38: -0.0891136 124: -0.00696326 126: -0.0156931 127: -0.0153151 128: 0.0734567
Row 129: 25: 0.0905411 35: -0.0683654 37: -0.102507 38: 0.0803318 106: -0.0171931 107: -0.00544221 129: 0.074479
Row 130: 28: 0.0859151 35: -0.117928 37: 0.100614 38: -0.0686011 113: -0.00871649 114: -0.0127623 129: -0.0084338 130: 0.0733096
Row 131: 29: 0.0897638 34: 0.0846056 36: -0.115247 38: -0.0591227 116: -0.00288065 117: -0.0195603 127: -0.0119 128: -0.00925136 131: 0.0757375
Row 132: 33: 0.104161 35: 0.113475 37: -0.160957 38: -0.0566793 125: -0.00252067 126: -0.0235195 129: -0.0116491 130: -0.0167197 132: 0.0772338
6. Solve the system:¶
[10]:
gfu.vec.data = \
a.mat.Inverse(freedofs=fes.FreeDofs()) * f.vec
Draw(gfu)
The Dirichlet boundary condition constrains some degrees of freedom. The argument fes.FreeDofs() indicates that only the remaining "free" degrees of freedom should participate in the linear solve.
You can examine the coefficient vector of solution if needed:
[11]:
print(gfu.vec)
0
0
0
0.0923019
0
0
0
0
0
0
0
0
0.0578968
0.086337
0.0954017
0.0944824
0.0888267
0.078059
0.0595638
0.0331316
0.0428238
0.0391783
0.0329405
0.0190228
0.0396205
0.0437688
0.0470417
0.0751353
0.0891838
0.0930866
0.0914133
0.0837669
0.0706131
0.0597334
0.0673558
0.0730976
0.0844105
0.0647765
0.0814921
0
-0.00563511
0
0
0.0213178
0
-0.0350904
0.00259849
-0.00728873
-0.00280466
0
-0.00967023
-0.0123906
0
-0.0240987
-0.0135412
0
-0.0171313
-0.0200732
-0.00676928
-0.0212666
0
-0.0254979
-0.00995785
0
-0.0390824
-0.013713
0
-0.0269058
-0.0176991
-0.0333808
-0.0238911
-0.0243047
-0.00442431
-0.0102753
-0.0146405
-0.006598
-0.00679179
-0.00570818
-0.00931926
-0.00609179
-0.00771503
-0.00532019
-0.00755367
0.00262541
-0.00112314
-0.00818072
0.00105745
0.000896631
-0.00820643
-0.00077952
0.00155068
0.00436689
-0.00490164
-0.01261
-0.00903107
-0.00751952
-0.0136094
-0.013109
-0.00783553
-0.0252117
-0.00819149
-0.00330874
-0.00919931
-0.0212406
-0.00570119
-0.00323099
-0.0135482
-0.0185787
-0.0142031
-0.0209225
-0.013664
-0.00405249
-0.00922214
-0.0103465
-0.00890607
-0.00277651
-0.00675828
-0.0105432
-0.00452866
-0.00654881
-0.00618555
-0.000547291
-0.00262347
-0.00559157
-0.00883386
-0.0116287
-0.010581
-0.00921447
-0.0126657
-0.00446274
-0.0171414
-0.00785433
-0.0118158
Ways to interact with NGSolve¶
A jupyter notebook (like this one) gives you one way to interact with NGSolve. When you have a complex sequence of tasks to perform, the notebook may not be adequate.
You can write an entire python script in a text editor and call python on the command line. (A script of the above is provided in
poisson.py.)python3 poisson.py
If you want the Netgen GUI, then use
netgenon the command line:netgen poisson.py
You can then ask for a python shell from the GUI’s menu options (
Solve -> Python shell).
[ ]: