This page was generated from wta/poisson.ipynb.
Poisson EquationΒΆ
Import Netgen/NGSolve Python modules:
[1]:
from ngsolve import *
# from netgen.geom2d import unit_square
from netgen.occ import unit_square
from ngsolve.webgui import Draw
The unit_square is a predefined domain, use Netgen to generate a mesh:
[2]:
mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))
Draw (mesh);
Define a finite element space on that mesh.
[3]:
fes = H1(mesh, order=3, dirichlet="left|right|bottom|top")
print ("ndof =", fes.ndof)
ndof = 283
Define linear and bilinear-forms.
\[a(u,v) = \int \nabla u \nabla v
\qquad \text{and} \qquad
f(v) = \int f v\]
Forms are expressed in terms of trial and test-functions:
[4]:
u = fes.TrialFunction()
v = fes.TestFunction()
f = LinearForm(fes)
f += 32*(y*(1-y)+x*(1-x))*v*dx
a = BilinearForm(fes)
a += grad(u)*grad(v)*dx
a.Assemble()
f.Assemble()
[4]:
<ngsolve.comp.LinearForm at 0x7f70f804f1b0>
[5]:
print(f.vec)
print(a.mat)
0.0196267
0.0318555
0.0196267
0.0432345
0.14621
0.175514
0.151105
0.0906392
0.107372
0.169991
0.156255
0.128848
0.180894
0.141079
0.129285
0.0951488
0.100589
0.154833
0.107972
0.248598
0.528919
0.437267
0.331342
0.167338
0.43029
0.416547
0.398158
0.305448
0.429221
0.268588
0.279976
0.326097
0.669264
0.433187
0.622777
0.619926
0.840647
0.334793
0.428205
-0.00291556
-0.000294603
-0.00291556
-0.000294603
-0.00219132
-0.000190646
-0.00225244
-0.000196066
-0.00500945
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-1.23197e-05
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2.3915e-06
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0.000125781
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1.66965e-07
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9.46593e-05
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Row 0: 0: 1 4: -0.5 19: -0.5 39: -0.0833333 40: -2.71484e-16 41: -0.0833333 42: -3.26128e-16 61: 0.166667 62: -4.85723e-17 227: -4.16334e-17
Row 1: 1: 0.829409 7: -0.184983 8: -0.193283 23: -0.451143 43: -0.0369847 44: -1.01048e-16 45: -0.0382058 46: -1.07987e-16 47: -0.0630443 48: -2.73219e-16 79: 0.0678152 80: -2.94903e-17 83: 0.0704196 84: -2.77556e-17 233: -2.42861e-17 235: -2.08167e-17
Row 2: 2: 1 11: -0.5 12: -0.5 49: -0.0833333 50: -2.68015e-16 51: -0.0833333 52: -3.26128e-16 99: 0.166667 100: -4.85723e-17 242: -5.89806e-17
Row 3: 3: 0.869547 15: -0.338672 16: -0.338125 30: -0.192751 53: -0.0161765 54: -1.34224e-16 55: -0.0159487 56: -1.30972e-16 57: -0.112799 58: -2.09034e-16 123: 0.0726218 124: 3.1225e-17 127: 0.0723028 128: 3.1225e-17 248: -1.21431e-17 250: -6.93889e-18
Row 4: 0: -0.5 4: 1.88285 5: -0.365191 19: -0.202546 20: -0.815113 39: 6.59195e-17 40: 1.77918e-16 41: 0.0833333 42: 9.36751e-17 59: -0.0499253 60: -1.79978e-16 61: -0.16926 62: -2.47198e-16 63: -0.0946228 64: -4.85723e-16 67: 0.11079 68: -1.9082e-17 141: 0.119684 142: -1.04083e-16 227: -3.46945e-18 228: -3.1225e-17 256: -5.20417e-17
Row 5: 4: -0.365191 5: 1.75257 6: -0.291174 20: -0.429447 21: -0.66676 59: -0.0352533 60: 1.64365e-16 63: 0.0961185 64: 3.46945e-18 65: -0.0490111 66: -1.50921e-16 67: -0.122981 68: -3.35669e-16 69: -0.0848502 70: -3.59955e-16 73: 0.0975401 74: -2.77556e-17 145: 0.0984366 146: -6.93889e-18 228: -2.08167e-17 229: -2.77556e-17 230: -2.08167e-17
Row 6: 5: -0.291174 6: 1.74123 7: -0.236328 21: -0.527617 22: -0.686112 65: -0.0468114 66: 1.44416e-16 69: 0.0953403 70: -3.1225e-17 71: -0.0556534 72: -1.47451e-16 73: -0.107228 74: -3.56486e-16 75: -0.0805129 76: -3.65593e-16 77: 0.0950414 78: -5.20417e-17 151: 0.0998236 152: 3.46945e-18 229: -2.60209e-17 231: -3.1225e-17 232: -2.08167e-17
Row 7: 1: -0.184983 6: -0.236328 7: 1.82052 22: -0.447625 23: -0.951582 43: -0.0855883 44: 1.5786e-16 47: 0.116419 48: -1.07553e-16 71: -0.0500031 72: 1.40513e-16 75: 0.0893912 76: -3.81639e-17 77: -0.112397 78: -3.07913e-16 79: -0.0554315 80: -4.97866e-16 157: 0.0976098 158: -9.71445e-17 231: -3.29597e-17 233: -4.85723e-17 234: -3.98986e-17
Row 8: 1: -0.193283 8: 1.86126 9: -0.342698 23: -1.02074 24: -0.304541 45: -0.0811378 46: 1.43982e-16 47: 0.113352 48: -1.00614e-16 81: -0.0368964 82: -1.63931e-16 83: -0.0460742 84: -5.18682e-16 85: -0.146101 86: -2.87964e-16 89: 0.0940127 90: 3.46945e-18 163: 0.102845 164: 1.31839e-16 235: -5.11743e-17 236: -1.73472e-17 237: -4.85723e-17
Row 9: 8: -0.342698 9: 1.74018 10: -0.284719 24: -0.618379 25: -0.494385 81: -0.0514511 82: 1.80411e-16 85: 0.108567 86: -1.73472e-17 87: -0.0356266 88: -1.39212e-16 89: -0.103887 90: -3.98986e-16 91: -0.0990653 92: -3.07046e-16 95: 0.0830798 96: -6.93889e-18 165: 0.0983831 166: 3.46945e-17 236: -3.46945e-17 238: -2.08167e-17 239: -3.1225e-17
Row 10: 9: -0.284719 10: 1.76227 11: -0.301882 25: -0.774714 26: -0.400953 87: -0.0632386 88: 1.66533e-16 91: 0.110692 92: -5.55112e-17 93: -0.0333683 94: -1.45283e-16 95: -0.0809105 96: -4.47559e-16 97: -0.116194 98: -2.98372e-16 101: 0.0836819 102: 1.73472e-18 171: 0.0993376 172: 7.80626e-17 238: -3.81639e-17 240: -2.08167e-17 241: -3.81639e-17
Row 11: 2: -0.5 10: -0.301882 11: 1.93492 12: -0.144458 26: -0.988582 49: -3.46945e-18 50: 1.76291e-16 51: 0.0833333 52: 9.28077e-17 93: -0.0629236 94: 1.75641e-16 97: 0.113237 98: -4.85723e-17 99: -0.185173 100: -2.48933e-16 101: -0.0743898 102: -5.39499e-16 105: 0.125916 106: -1.42247e-16 240: -4.51028e-17 242: 3.46945e-18 257: -5.55112e-17
Row 12: 2: -0.5 11: -0.144458 12: 1.80641 13: -0.248578 26: -0.240805 27: -0.672566 49: 0.0833333 50: 9.19403e-17 51: -7.63278e-17 52: 2.30718e-16 99: -0.119012 100: 2.32887e-16 101: 0.0597549 102: -3.46945e-17 103: -0.0383155 104: -1.27068e-16 105: -0.0978551 106: -2.10769e-16 107: -0.0458853 108: -3.1019e-16 111: 0.0797452 112: -1.73472e-17 177: 0.0782344 178: -6.93889e-17 242: 5.42101e-17 243: -2.08167e-17 257: -2.47198e-17 258: -6.93889e-18
Row 13: 12: -0.248578 13: 1.75348 14: -0.262981 27: -0.741803 28: -0.500119 103: -0.067177 104: 1.58727e-16 107: 0.108607 108: -5.89806e-17 109: -0.0493328 110: -1.48753e-16 111: -0.0754501 112: -4.23273e-16 113: -0.100287 114: -3.10516e-16 117: 0.093163 118: -2.94903e-17 181: 0.0904772 182: 5.89806e-17 243: -4.59702e-17 244: -3.1225e-17 245: -3.81639e-17
Row 14: 13: -0.262981 14: 1.79883 15: -0.166757 28: -0.478926 29: -0.890168 109: -0.0513313 110: 1.52656e-16 113: 0.0951615 114: -3.46945e-17 115: -0.0772174 116: -1.43115e-16 117: -0.114974 118: -3.57353e-16 119: -0.0562824 120: -4.24574e-16 121: 0.10501 122: -9.19403e-17 185: 0.0996335 186: -3.1225e-17 244: -3.29597e-17 246: -4.51028e-17 247: -6.93889e-18
Row 15: 3: -0.338672 14: -0.166757 15: 1.78899 29: -0.48172 30: -0.801839 53: -0.0830516 54: 2.00361e-16 57: 0.139497 58: -8.32667e-17 115: -0.0456942 116: 1.09288e-16 119: 0.073487 120: -4.33681e-17 121: -0.0783809 122: -2.81025e-16 123: -0.0910377 124: -4.64906e-16 191: 0.0851806 192: -5.20417e-17 246: -2.77556e-17 248: -4.85723e-17 249: -2.68882e-17
Row 16: 3: -0.338125 16: 1.79703 17: -0.209944 30: -0.859546 31: -0.389415 55: -0.083616 56: 2.02095e-16 57: 0.13997 58: -7.97973e-17 125: -0.0366943 126: -1.14925e-16 127: -0.0845623 128: -4.77049e-16 129: -0.0946323 130: -2.61943e-16 133: 0.0716849 134: -2.60209e-17 195: 0.0878499 196: 6.59195e-17 250: -5.29091e-17 251: -2.08167e-17 252: -3.81639e-17
Row 17: 16: -0.209944 17: 1.91346 18: -0.373491 31: -1.10417 32: -0.225859 125: -0.0789635 126: 1.62197e-16 129: 0.113954 130: -8.67362e-17 131: -0.0159351 132: -1.40513e-16 133: -0.0566987 134: -5.72459e-16 135: -0.167313 136: -2.61943e-16 139: 0.0781836 140: 3.98986e-17 199: 0.126773 200: 1.49186e-16 251: -5.1608e-17 253: -6.93889e-18 254: -6.59195e-17
Row 18: 17: -0.373491 18: 2.04304 19: -0.564712 32: -1.10484 131: -0.076135 132: 2.04697e-16 135: 0.138383 136: -5.89806e-17 137: -0.108005 138: -3.17454e-16 139: -0.156367 140: -6.74807e-16 143: 0.202124 144: -7.97973e-17 253: -5.20417e-17 255: -6.245e-17
Row 19: 0: -0.5 4: -0.202546 18: -0.564712 19: 1.87677 20: -0.483092 32: -0.126419 39: 0.0833333 40: 8.93383e-17 41: -1.38778e-17 42: 2.27249e-16 61: -0.11712 62: 2.40259e-16 63: 0.0675445 64: -2.1684e-17 137: 0.0159351 138: 1.81279e-16 139: 0.0781836 140: 1.31839e-16 141: -0.0707626 142: -2.98372e-16 143: -0.140847 144: -1.89952e-16 147: 0.0837334 148: 3.98986e-17 227: 4.51028e-17 255: 3.46945e-18 256: -2.25514e-17 259: -3.55618e-17
Row 20: 4: -0.815113 5: -0.429447 19: -0.483092 20: 3.55106 21: -0.571322 32: -0.548291 34: -0.7038 59: 0.0851786 60: 1.47451e-17 61: 0.119714 62: 6.15827e-17 63: -0.0690402 64: 4.95264e-16 67: -0.0975533 68: 3.6169e-16 69: 0.0839491 70: -1.30104e-17 141: -0.148211 142: 4.70977e-16 143: 0.109013 144: -1.30104e-17 145: -0.0860313 146: -3.37404e-16 147: -0.114318 148: -3.76435e-16 149: -0.0766901 150: -4.0766e-16 155: 0.0973024 156: -3.1225e-17 203: 0.0966876 204: -7.80626e-17 228: 5.37764e-17 230: 5.55112e-17 256: 7.28584e-17 259: 6.59195e-17 261: -3.46945e-17 272: -4.59702e-17
Row 21: 5: -0.66676 6: -0.527617 20: -0.571322 21: 3.48835 22: -0.5263 33: -0.710873 34: -0.485478 65: 0.0958225 66: 5.20417e-18 67: 0.109743 68: -6.93889e-18 69: -0.0944393 70: 4.07226e-16 73: -0.0943959 74: 4.0289e-16 75: 0.0865096 76: -5.20417e-18 145: -0.109171 146: 3.49547e-16 149: 0.0946479 150: -2.77556e-17 151: -0.0956413 152: -3.46511e-16 153: -0.0761893 154: -3.72098e-16 155: -0.111555 156: -3.46945e-16 161: 0.0968484 162: -4.51028e-17 207: 0.0978196 208: 1.56125e-17 229: 5.55112e-17 230: -3.46945e-17 232: 5.37764e-17 260: -2.77556e-17 261: -3.46945e-17 262: -2.08167e-17
Row 22: 6: -0.686112 7: -0.447625 21: -0.5263 22: 3.50536 23: -0.714039 24: -0.512751 33: -0.618534 71: 0.105656 72: 9.54098e-18 73: 0.104083 74: -1.04083e-17 75: -0.0953879 76: 4.11997e-16 77: -0.108398 78: 3.44776e-16 79: 0.0773453 80: -5.89806e-17 151: -0.106574 152: 3.34368e-16 153: 0.0902074 154: -3.46945e-17 157: -0.0594121 158: -4.09395e-16 159: -0.121476 160: -3.71231e-16 161: -0.0929793 162: -3.7817e-16 163: 0.101073 164: 3.46945e-17 167: 0.105861 168: -3.64292e-17 231: 6.59195e-17 232: -3.46945e-17 234: -3.81639e-17 260: -2.77556e-17 265: 6.245e-17 266: -3.1225e-17
Row 23: 1: -0.451143 7: -0.951582 8: -1.02074 22: -0.714039 23: 3.746 24: -0.608496 43: 0.122573 44: -5.55112e-17 45: 0.119344 46: -3.98986e-17 47: -0.166726 48: 4.78784e-16 77: 0.125753 78: 1.73472e-17 79: -0.0897289 80: 5.88071e-16 83: -0.0937365 84: 6.12357e-16 85: 0.144516 86: 3.46945e-17 157: -0.118894 158: 4.96565e-16 159: 0.112148 160: 5.89806e-17 163: -0.155247 164: -4.64039e-16 233: 7.63278e-17 234: 7.97973e-17 235: 7.28584e-17 237: 8.67362e-17 265: -3.81639e-17
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Row 25: 9: -0.494385 10: -0.774714 24: -0.425534 25: 3.61904 26: -0.637739 35: -0.287882 37: -0.998787 87: 0.0988652 88: -2.51535e-17 89: 0.0928205 90: -2.42861e-17 91: -0.109288 92: 3.55618e-16 95: -0.0833453 96: 4.93529e-16 97: 0.113599 98: 2.1684e-17 165: -0.125908 166: 3.07046e-16 169: 0.10401 170: -1.09288e-16 171: -0.09982 172: -3.99854e-16 173: -0.145898 174: -2.76688e-16 175: -0.0389135 176: -4.96131e-16 179: 0.0925107 180: 1.04083e-17 219: 0.101368 220: -1.31839e-16 238: 5.89806e-17 239: -3.64292e-17 241: 6.59195e-17 263: -2.08167e-17 274: -5.50775e-17 277: -5.20417e-17
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Row 164: 8: 1.31839e-16 22: 3.46945e-17 23: -4.64039e-16 24: 3.00541e-16 83: 0.00514227 84: 0.000462015 85: -0.00514227 86: -0.00296616 157: 0.00505367 158: 0.00116037 159: -0.00505367 160: -0.00220874 163: 0.00532879 164: 0.0237362 237: -0.00250415 265: -0.00104837
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Row 172: 10: 7.80626e-17 25: -3.99854e-16 26: 3.46077e-16 35: -1.82146e-17 95: 0.00496688 96: 0.00111524 97: -0.00496688 98: -0.00219601 171: 0.000639222 172: 0.023086 173: 0.0043759 174: -0.00178594 179: -0.0043759 180: 0.00113132 241: -0.00108077 263: 0.000654621
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Row 174: 25: -2.76688e-16 26: -2.77556e-17 35: 2.69533e-16 37: 3.1225e-17 171: 0.00560747 172: -0.00178594 173: 0.00179542 174: 0.0218572 175: 0.00718693 176: -0.00188038 179: -0.00560747 180: -0.00195237 219: -0.00718693 220: 0.0029109 263: -0.000166424 277: -0.00103052
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Row 177: 12: 0.0782344 26: -0.0433393 27: -0.123366 35: 0.0884711 105: -0.0184447 106: -0.00791708 107: -0.0011139 108: -0.00445092 177: 0.076992 178: -0.00400134 179: -0.0123969 180: 0.00531436 183: -0.00972093 184: 0.00477918 258: -0.00845627 264: -0.00566999
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Row 182: 13: 5.89806e-17 27: -4.7011e-16 28: 3.76001e-16 35: 3.98986e-17 111: 0.00452386 112: 0.00113401 113: -0.00452386 114: -0.00188189 181: 0.00293876 182: 0.0243494 183: 0.00378432 184: -0.000655792 187: -0.00378432 188: 0.00186709 245: -0.00074788 270: -0.0012113
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Row 192: 15: -5.20417e-17 29: -4.96131e-16 30: 4.87674e-16 36: 5.72459e-17 121: 0.00425903 122: 0.00168627 123: -0.00425903 124: -0.00115308 191: 0.00185948 192: 0.0267831 193: 0.00307436 194: -0.000138366 197: -0.00307436 198: 0.00191121 249: 0.000533191 267: -0.00177284
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Row 194: 28: 3.81639e-17 29: -4.84855e-16 30: 1.14492e-16 36: 3.30465e-16 185: 0.00550686 186: 0.000933189 189: -0.00550686 190: -0.00273805 191: 0.00566107 192: -0.000138366 193: 0.00795327 194: 0.0248415 197: -0.00566107 198: -0.00363568 267: -0.00349732 269: -0.00180486
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Row 196: 16: 6.59195e-17 30: -4.67508e-16 31: 4.59702e-16 36: -5.63785e-17 127: 0.0043925 128: 0.000940274 129: -0.0043925 130: -0.00198806 195: -0.00119266 196: 0.0262751 197: 0.00330041 198: -0.00202158 201: -0.00330041 202: 0.00017869 252: -0.00104778 268: 0.00184289
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Row 274: 24: -2.77556e-17 25: -5.50775e-17 37: 9.02056e-17 165: -0.00248728 166: 0.00104318 169: -0.00478422 170: 0.00180882 175: -0.00791374 176: 0.000765645 274: 0.0101235
Row 275: 24: -2.08167e-17 33: 6.76542e-17 37: -5.20417e-17 167: -0.00264163 168: 0.001157 169: -0.00453548 170: 0.00178828 209: -0.00800647 210: 0.000631285 275: 0.0101224
Row 276: 28: -3.46945e-17 35: -1.38778e-17 36: 5.20417e-17 187: -0.00621435 188: -0.00120278 189: -0.00614054 190: -0.00122739 217: -0.00253219 218: -2.46043e-05 276: 0.00992472
Row 277: 25: -5.20417e-17 35: -3.81639e-17 37: 9.02056e-17 173: -0.00140407 174: -0.00103052 175: -0.00873271 176: 0.00141236 219: -0.00564115 220: 0.00244288 277: 0.0105186
Row 278: 33: -1.73472e-17 37: -4.68375e-17 38: 6.93889e-17 209: -0.00537389 210: 0.00101154 211: -0.00314003 212: 0.000266916 225: -0.00617463 226: -0.000744619 278: 0.00979236
Row 279: 33: -4.29344e-17 34: -3.81639e-17 38: 7.28584e-17 207: -0.00307396 208: -4.34468e-05 211: -0.00587366 212: 0.000889789 215: -0.00574332 216: 0.000933236 279: 0.00979396
Row 280: 35: -2.34188e-17 37: -4.85723e-17 38: 7.28584e-17 219: -0.00324107 220: 0.00127552 221: -0.00398948 222: 0.00152499 225: -0.00781603 226: 0.00024947 280: 0.0100311
Row 281: 34: -6.41848e-17 36: -2.60209e-17 38: 9.36751e-17 213: -0.000875849 214: -0.00168221 215: -0.0103984 216: 0.00149197 223: -0.00535176 224: 0.00317418 281: 0.011084
Row 282: 35: -5.55112e-17 36: -3.1225e-17 38: 8.67362e-17 217: -0.00142194 218: -0.00138971 221: -0.00936961 222: 0.00125952 223: -0.00520049 224: 0.00264922 282: 0.0106614
Solve the problem:
[6]:
gfu = GridFunction(fes)
gfu.vec.data = a.mat.Inverse(fes.FreeDofs()) * f.vec
Plot the solution:
[7]:
Draw (gfu, mesh);
[8]:
Draw (-grad(gfu), mesh, "Flux", vectors= { "grid_size" : 40});
Calculate error:
[9]:
exact = 16*x*(1-x)*y*(1-y)
print ("L2-error:", sqrt(Integrate((gfu-exact)**2, mesh)))
L2-error: 5.5844587523451645e-05
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