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 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)
[2]:
BaseWebGuiScene
Define a finite element space on that mesh.
[3]:
fes = H1(mesh, order=3, dirichlet="left|right|bottom|top")
print ("ndof =", fes.ndof)
ndof = 274
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 0x7f159d1e52b0>
[5]:
print(f.vec)
print(a.mat)
0.0196267
0.0333189
0.0196267
0.0463297
0.141964
0.177229
0.160503
0.0980466
0.11023
0.167123
0.147227
0.121155
0.175997
0.174503
0.172725
0.11678
0.0613188
0.298698
0.103856
0.222382
0.485193
0.458902
0.363518
0.179717
0.456656
0.390526
0.357482
0.309012
0.606224
0.319488
0.32271
0.538616
0.51025
0.610927
0.454717
0.756316
0.370579
0.607194
-0.00291556
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Row 0: 0: 1 4: -0.5 19: -0.5 38: -0.0833333 39: -2.61943e-16 40: -0.0833333 41: -3.19189e-16 60: 0.166667 61: -4.16334e-17 220: -4.85723e-17
Row 1: 1: 0.828444 7: -0.208707 8: -0.210132 23: -0.409604 42: -0.034013 43: -1.33574e-16 44: -0.0342543 45: -1.04083e-16 46: -0.0698066 47: -2.43295e-16 78: 0.0687976 79: 2.34188e-17 82: 0.0692764 83: -2.42861e-17 226: -1.73472e-17 228: -1.9082e-17
Row 2: 2: 1 11: -0.5 12: -0.5 48: -0.0833333 49: -3.19189e-16 50: -0.0833333 51: -2.70617e-16 98: 0.166667 99: 4.16334e-17 235: -4.85723e-17
Row 3: 3: 0.907169 15: -0.407338 16: -0.376463 30: -0.123368 52: -0.020713 53: -2.06432e-16 54: 0.000151649 55: -1.36881e-16 56: -0.130634 57: -1.83013e-16 122: 0.0886026 123: 8.67362e-17 126: 0.0625923 127: 6.85216e-17 241: -4.16334e-17 243: 1.73472e-18
Row 4: 0: -0.5 4: 1.8918 5: -0.369703 19: -0.170695 20: -0.851406 38: 0 39: 1.76942e-16 40: 0.0833333 41: 9.71445e-17 58: -0.0553087 59: -1.9082e-16 60: -0.169926 61: -2.6628e-16 62: -0.0900664 63: -4.46691e-16 66: 0.116926 67: -2.42861e-17 140: 0.115041 141: -4.51028e-17 220: 0 221: -3.46945e-17 249: -1.38778e-17
Row 5: 4: -0.369703 5: 1.76479 6: -0.315355 20: -0.373272 21: -0.70646 58: -0.0302454 59: 1.66533e-16 62: 0.0918626 63: 1.73472e-17 64: -0.0537908 65: -1.70003e-16 66: -0.12557 67: -3.10082e-16 68: -0.0845258 69: -3.75568e-16 72: 0.10635 73: -2.77556e-17 144: 0.0959192 145: -2.08167e-17 221: -2.60209e-17 222: -3.46945e-17 223: -1.38778e-17
Row 6: 5: -0.315355 6: 1.74934 7: -0.271622 21: -0.444807 22: -0.71756 64: -0.0387141 65: 1.53957e-16 68: 0.0912733 69: -3.46945e-18 70: -0.0577312 71: -1.63064e-16 72: -0.114421 73: -3.322e-16 74: -0.0806907 75: -3.77736e-16 76: 0.103002 77: -4.51028e-17 150: 0.0972825 151: -6.93889e-18 222: -2.77556e-17 224: -3.81639e-17 225: -2.08167e-17
Row 7: 1: -0.208707 6: -0.271622 7: 1.82575 22: -0.375475 23: -0.969947 42: -0.0837431 43: 2.35055e-16 46: 0.118528 47: -4.94396e-17 70: -0.0420473 71: 1.43982e-16 74: 0.0873176 75: -1.73472e-17 76: -0.123185 77: -3.03577e-16 78: -0.0553164 79: -4.64039e-16 156: 0.0984466 157: -5.29091e-17 224: -3.1225e-17 226: 8.32667e-17 227: -1.38778e-17
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Row 9: 8: -0.354044 9: 1.7442 10: -0.268795 24: -0.640588 25: -0.480773 80: -0.0552752 81: 1.85182e-16 84: 0.114283 85: -2.77556e-17 86: -0.0341102 87: -1.28803e-16 88: -0.105026 89: -4.15466e-16 90: -0.0962885 91: -2.90566e-16 94: 0.0789093 95: -5.20417e-18 164: 0.0975081 165: 3.81639e-17 229: -3.64292e-17 231: -2.08167e-17 232: -2.08167e-17
Row 10: 9: -0.268795 10: 1.77359 11: -0.272339 25: -0.832083 26: -0.400373 86: -0.0686401 87: 1.63064e-16 90: 0.113439 91: -6.245e-17 92: -0.0327251 93: -1.26635e-16 94: -0.0788029 95: -4.54498e-16 96: -0.11543 97: -2.84495e-16 100: 0.078115 101: -7.80626e-18 170: 0.104044 171: 7.97973e-17 231: -4.51028e-17 233: -1.73472e-17 234: -4.51028e-17
Row 11: 2: -0.5 10: -0.272339 11: 1.97625 12: -0.118504 26: -1.0854 48: -6.93889e-17 49: 2.28983e-16 50: 0.0833333 51: 8.67362e-17 92: -0.0698849 93: 1.68268e-16 96: 0.115275 97: -6.245e-17 98: -0.194349 99: -2.9577e-16 100: -0.0651405 101: -5.6205e-16 104: 0.130766 105: -1.66533e-16 233: -3.64292e-17 235: 5.20417e-17 250: -6.245e-17
Row 12: 2: -0.5 11: -0.118504 12: 1.80573 13: -0.26228 26: -0.248259 27: -0.676685 48: 0.0833333 49: 9.36751e-17 50: 3.46945e-18 51: 1.8323e-16 98: -0.119671 99: 1.71304e-16 100: 0.0560888 101: -3.98986e-17 102: -0.0411842 103: -1.36609e-16 104: -0.0913473 105: -2.10769e-16 106: -0.0487517 107: -3.19406e-16 110: 0.0848976 111: -2.08167e-17 176: 0.0766349 177: -6.59195e-17 235: -3.46945e-18 236: -2.77556e-17 250: -2.29851e-17 251: 3.46945e-18
Row 13: 12: -0.26228 13: 1.81175 14: -0.428891 27: -0.813095 28: -0.307486 102: -0.0605924 103: 1.56992e-16 106: 0.104306 107: -4.85723e-17 108: -0.0361696 109: -1.93422e-16 110: -0.0587915 111: -4.4062e-16 112: -0.146405 113: -2.91434e-16 116: 0.107651 117: 1.73472e-17 180: 0.0900017 181: 1.05818e-16 236: -3.46945e-17 237: -2.08167e-17 238: -3.81639e-17
Row 14: 13: -0.428891 14: 1.79855 15: -0.254586 28: -0.343721 29: -0.771353 108: -0.0404916 109: 1.97325e-16 112: 0.111973 113: 1.38778e-17 114: -0.0578574 115: -1.56992e-16 116: -0.142183 117: -3.08781e-16 118: -0.0592263 119: -3.70147e-16 120: 0.100288 121: -4.85723e-17 184: 0.0874967 185: -4.77049e-17 237: -2.42861e-17 239: -3.46945e-17 240: -1.38778e-17
Row 15: 3: -0.407338 14: -0.254586 15: 1.75967 29: -0.52416 30: -0.573581 52: -0.0625066 53: 2.90566e-16 56: 0.130396 57: 1.73472e-18 114: -0.0447659 115: 1.37911e-16 118: 0.0871969 119: -2.08167e-17 120: -0.0755213 121: -3.27863e-16 122: -0.110484 123: -3.98986e-16 192: 0.0756844 193: 3.81639e-17 239: -3.20924e-17 241: 7.97973e-17 242: -2.42861e-17
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Row 17: 16: -0.374644 17: 1.78398 18: -0.43722 30: -0.336404 31: -0.30274 35: -0.332973 124: -0.000151649 125: 1.27177e-16 126: 0.0625923 127: 6.76542e-17 128: -0.000386213 129: -1.55258e-16 130: -0.0860908 131: -2.48065e-16 132: -0.104715 133: -2.2855e-16 134: -0.105986 135: -2.55004e-16 138: 0.0732563 139: 8.06646e-17 196: 0.0795659 197: 5.89806e-17 200: 0.0819158 201: -8.67362e-19 245: -1.73472e-18 246: 1.73472e-18 261: -3.46945e-17 264: -1.38778e-17
Row 18: 17: -0.43722 18: 2.01666 19: -0.441855 31: -1.13758 128: -0.0944124 129: 2.55004e-16 132: 0.167282 133: -6.93889e-17 136: -0.0951848 137: -2.61943e-16 138: -0.146513 139: -6.9042e-16 142: 0.168827 143: -7.63278e-17 246: -5.55112e-17 248: -6.245e-17
Row 19: 0: -0.5 4: -0.170695 18: -0.441855 19: 1.80929 20: -0.457008 31: -0.239727 38: 0.0833333 39: 8.67362e-17 40: -7.63278e-17 41: 2.28983e-16 60: -0.122284 61: 2.78857e-16 62: 0.0674001 63: -6.07153e-18 136: 0.000386213 137: 1.59811e-16 138: 0.0732563 139: 7.97973e-17 140: -0.0687899 141: -2.77989e-16 142: -0.11086 143: -2.34188e-16 146: 0.0775578 147: -2.94903e-17 220: 5.20417e-17 248: -3.46945e-18 249: 3.98986e-17 252: -3.1225e-17
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Row 21: 5: -0.70646 6: -0.444807 20: -0.621608 21: 3.49846 22: -0.562802 32: -0.707916 33: -0.454863 64: 0.0925049 65: 1.64799e-17 66: 0.115038 67: -1.38778e-17 68: -0.0897997 69: 4.02456e-16 72: -0.102651 73: 3.71665e-16 74: 0.0842808 75: -1.47451e-17 144: -0.106796 145: 3.68195e-16 148: 0.095359 149: -1.38778e-17 150: -0.0895218 151: -3.38705e-16 152: -0.0779072 153: -3.70797e-16 154: -0.1164 155: -3.39138e-16 160: 0.0990414 161: -3.98986e-17 202: 0.0968518 203: 1.38778e-17 222: 6.07153e-17 223: -2.94903e-17 225: 5.89806e-17 253: -2.08167e-17 254: -2.94903e-17 255: -2.08167e-17
Row 22: 6: -0.71756 7: -0.375475 21: -0.562802 22: 3.51385 23: -0.705204 24: -0.539682 32: -0.613124 70: 0.0997784 71: 1.73472e-17 72: 0.110723 73: -1.38778e-17 74: -0.0909077 75: 4.05058e-16 76: -0.112022 77: 3.51715e-16 78: 0.0748223 79: -3.55618e-17 150: -0.10743 151: 3.34802e-16 152: 0.090508 153: -3.1225e-17 156: -0.055892 157: -3.46511e-16 158: -0.119863 159: -3.34802e-16 160: -0.0995268 161: -3.88578e-16 162: 0.0986036 163: 1.04083e-17 166: 0.111206 167: -3.46945e-17 224: 6.59195e-17 225: -3.46945e-17 227: 4.85723e-17 253: -3.29597e-17 258: -2.08167e-17 259: -3.1225e-17
Row 23: 1: -0.409604 7: -0.969947 8: -1.05651 22: -0.705204 23: 3.76626 24: -0.624994 42: 0.117756 43: -1.00614e-16 44: 0.11718 45: -4.77049e-17 46: -0.166669 47: 3.96384e-16 76: 0.132205 77: 0 78: -0.0883035 79: 4.8312e-16 82: -0.0906718 83: 6.36644e-16 84: 0.149578 85: 4.16334e-17 156: -0.125663 157: 4.04191e-16 158: 0.110992 159: -1.38778e-17 162: -0.156404 163: -5.05672e-16 226: -6.20164e-17 227: -3.1225e-17 228: 6.93889e-17 230: 9.02056e-17 258: -2.60209e-17
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Row 161: 21: -3.98986e-17 22: -3.88578e-16 24: 0 32: 4.35416e-16 150: 0.00495207 151: 0.00180338 152: -0.00495207 153: -0.001498 158: 0.00446206 159: -0.00115515 160: 0.000538548 161: 0.0227709 166: -0.00446206 167: -0.00181956 253: 0.000305385 259: -0.000664417
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Row 163: 8: 1.42247e-16 22: 1.04083e-17 23: -5.05672e-16 24: 3.60822e-16 82: 0.00522246 83: 0.0003763 84: -0.00522246 85: -0.00310534 156: 0.00493018 157: 0.00117867 158: -0.00493018 159: -0.00210812 162: 0.00548773 163: 0.0239534 230: -0.00272904 258: -0.000929446
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Row 165: 9: 3.81639e-17 24: -3.17888e-16 25: 3.20924e-16 36: -3.38271e-17 88: 0.00487541 89: 0.00153396 90: -0.00487541 91: -0.00171632 164: -0.00144033 165: 0.0222402 168: 0.00591439 169: -0.00254275 174: -0.00591439 175: 0.00140018 232: -0.000182359 267: 0.00114258
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Row 167: 22: -3.46945e-17 24: -2.71918e-16 32: 3.83374e-16 36: -7.63278e-17 158: 0.00556031 159: 0.00188731 160: -0.00556031 161: -0.00181956 166: -0.00162001 167: 0.021816 168: 0.0064571 169: -0.0026585 204: -0.0064571 205: 0.00164624 259: 6.77457e-05 268: 0.00101226
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Row 171: 10: 7.97973e-17 25: -3.63858e-16 26: 3.08781e-16 34: -1.82146e-17 94: 0.00520221 95: 0.00113346 96: -0.00520221 97: -0.00233468 170: 0.000791927 171: 0.022507 172: 0.00504945 173: -0.00201978 178: -0.00504945 179: 0.00134652 234: -0.00120122 256: 0.000673268
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Row 173: 25: -2.78423e-16 26: -5.55112e-17 34: 3.21358e-16 36: 1.73472e-17 170: 0.00525603 171: -0.00201978 172: -0.000275032 173: 0.0218286 174: 0.00637567 175: -0.00194913 178: -0.00525603 179: -0.00148424 212: -0.00637567 213: 0.00230132 256: 0.000535548 269: -0.000352194
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Row 175: 24: -3.46945e-17 25: -4.52763e-16 34: -7.63278e-17 36: 5.6552e-16 164: 0.00368123 165: 0.00140018 168: -0.00368123 169: -0.00105397 172: 0.00406374 173: -0.00194913 174: -0.00230294 175: 0.0248588 212: -0.00406374 213: -0.000760034 267: 0.000346201 269: 0.00118909
Row 176: 12: 0.0766349 26: -0.0517256 27: -0.122692 34: 0.0977827 104: -0.0178992 105: -0.00787534 106: -0.00125957 107: -0.00454742 176: 0.0767377 177: -0.00354833 178: -0.0127739 179: 0.00488593 182: -0.0116718 183: 0.00466552 251: -0.00859102 257: -0.00466231
Row 177: 12: -6.59195e-17 26: -3.89012e-16 27: 4.44089e-16 34: -3.46945e-18 104: 0.00383175 105: 0.00238655 106: -0.00383175 107: -0.000167943 176: -0.00354833 177: 0.0248812 178: 0.00488914 179: -0.00170318 182: -0.00488914 183: 0.00155624 251: 0.00221861 257: 0.000146938
Row 178: 25: 0.0849225 26: -0.0911502 27: 0.0977186 34: -0.0914909 170: -0.0100989 171: -0.00504945 172: -0.0111318 173: -0.00525603 176: -0.0127739 177: 0.00488914 178: 0.0724805 179: -1.70376e-05 182: -0.0116558 183: -0.00466552 256: -0.00605935 257: -0.00466873
Row 179: 25: -2.25514e-17 26: -3.92048e-16 27: -3.1225e-17 34: 4.51028e-16 170: 0.00424613 171: 0.00134652 172: -0.00424613 173: -0.00148424 176: 0.00488593 177: -0.00170318 178: -1.70376e-05 179: 0.0229042 182: -0.00488593 183: -0.0015541 256: -0.00013772 257: 0.000149078
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Row 181: 13: 1.05818e-16 27: -4.8312e-16 28: 3.33284e-16 34: 3.90313e-17 110: 0.00450008 111: 0.000502604 112: -0.00450008 113: -0.00249745 180: 0.00478847 181: 0.0240909 182: 0.0047218 183: -0.000975203 186: -0.0047218 187: 0.00217267 238: -0.00199485 263: -0.00119746
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Row 183: 26: -2.42861e-17 27: -4.26742e-16 28: 1.04083e-17 34: 4.51895e-16 176: 0.00466552 177: 0.00155624 178: -0.00466552 179: -0.0015541 180: 0.00412997 181: -0.000975203 182: 0.00120115 183: 0.0233549 186: -0.00412997 187: -0.00177811 257: 2.13961e-06 263: -0.000802907
Row 184: 14: 0.0874967 28: -0.0325616 29: -0.127439 35: 0.0725036 116: -0.0176754 117: -0.00682491 118: -0.00419882 119: -0.0041296 184: 0.07731 185: -0.00474386 188: -0.0141843 189: 0.00700901 194: -0.00394157 195: 0.00496046 240: -0.00657967 262: -0.00834429
Row 185: 14: -4.77049e-17 28: -3.98119e-16 29: 4.90059e-16 35: -4.59702e-17 116: 0.00437483 117: 0.00235671 118: -0.00437483 119: -0.000559842 184: -0.00474386 185: 0.0255906 188: 0.00362518 189: -0.00189124 194: -0.00362518 195: 0.000525543 240: 0.00179687 262: 0.0013657
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Row 187: 27: -4.51028e-17 28: -3.13118e-16 34: 4.39752e-16 37: -7.97973e-17 180: 0.00592616 181: 0.00217267 182: -0.00592616 183: -0.00177811 186: -0.00241029 187: 0.0223306 190: 0.00490508 191: -0.00224118 214: -0.00490508 215: 0.00102888 263: 0.000394556 273: 0.0012123
Row 188: 28: -0.137443 29: 0.14018 35: -0.117127 37: 0.114391 184: -0.0141843 185: 0.00362518 188: 0.0756342 189: 0.0010158 190: -0.0150976 191: 0.00462949 194: -0.0208607 195: -0.00496046 216: -0.0135001 217: 0.00431001 247: -0.00321997 262: -0.00157663
Row 189: 28: -2.66714e-16 29: -6.93889e-18 35: 2.70183e-16 37: -6.93889e-18 184: 0.00700901 185: -0.00189124 188: 0.0010158 189: 0.021768 190: 0.00571953 191: -0.00201301 194: -0.00700901 195: -0.00278143 216: -0.00571953 217: 0.00180002 247: 0.000212991 262: -0.000890188
Row 190: 28: -0.0818333 34: 0.116869 35: 0.0925899 37: -0.127626 186: -0.0168088 187: 0.00490508 188: -0.0150976 189: 0.00571953 190: 0.0735814 191: -0.00228961 214: -0.0124084 215: -0.00402499 216: -0.00804992 217: -0.00431001 247: -0.00540005 273: -0.00308663
Row 191: 28: -3.01408e-16 34: -1.56125e-17 35: -6.41848e-17 37: 3.83374e-16 186: 0.00584345 187: -0.00224118 188: 0.00462949 189: -0.00201301 190: -0.00228961 191: 0.0224506 214: -0.00584345 215: -0.00165445 216: -0.00462949 217: -0.00107332 247: 0.000939684 273: 0.000586725
Row 192: 15: 0.0756844 29: -0.0798592 30: -0.0621923 35: 0.0663671 120: -0.00827257 121: -0.00531114 122: -0.0106485 123: -0.00578633 192: 0.07587 193: 0.000883346 194: -0.00727552 195: 0.0058699 196: -0.00931627 197: 0.00627805 242: -0.00731325 260: -0.0088296
Row 193: 15: 3.81639e-17 29: -5.1131e-16 30: 4.571e-16 35: 1.04083e-17 120: 0.00378422 121: 0.00110301 122: -0.00378422 123: -0.00141981 192: 0.000883346 193: 0.025613 194: 0.00331836 195: -0.000970069 196: -0.00331836 197: 0.00124217 242: -0.000316797 260: -0.000272101
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Row 195: 28: 6.15827e-17 29: -4.51895e-16 30: 5.20417e-17 35: 3.41741e-16 184: 0.00496046 185: 0.000525543 188: -0.00496046 189: -0.00278143 192: 0.0058699 193: -0.000970069 194: 0.00634353 195: 0.0238407 196: -0.0058699 197: -0.0029432 260: -0.00197313 262: -0.00225589
Row 196: 17: 0.0795659 29: 0.125561 30: -0.144212 35: -0.0609153 130: -0.00591256 131: -0.00471545 134: -0.0139789 135: -0.00632872 192: -0.00931627 193: -0.00331836 194: -0.022074 195: -0.0058699 196: 0.0762219 197: 0.00416482 260: -0.00291021 261: -0.00706587
Row 197: 17: 5.89806e-17 29: 3.29597e-17 30: -4.57967e-16 35: 3.66027e-16 130: 0.0039783 131: 0.000788341 134: -0.0039783 135: -0.00186386 192: 0.00627805 193: 0.00124217 194: -0.00627805 195: -0.0029432 196: 0.00416482 197: 0.0236512 260: -0.00170103 261: -0.00107552
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Solve the problem:
[6]:
gfu = GridFunction(fes)
gfu.vec.data = a.mat.Inverse(fes.FreeDofs()) * f.vec
Plot the solution:
[7]:
Draw (gfu, mesh)
[7]:
BaseWebGuiScene
[8]:
Draw (-grad(gfu), mesh, "Flux", vectors= { "grid_size" : 40})
[8]:
BaseWebGuiScene
Calculate error:
[9]:
exact = 16*x*(1-x)*y*(1-y)
print ("L2-error:", sqrt(Integrate((gfu-exact)**2, mesh)))
L2-error: 5.453884093826891e-05
[ ]: