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]:
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 0x7f35100609b0>
[5]:
print(f.vec)
print(a.mat)
0.0196267
0.0333126
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0.0433846
0.146631
0.184224
0.16284
0.0983461
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0.120123
0.173598
0.167877
0.160981
0.101885
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0.106049
0.237564
0.521936
0.471778
0.3658
0.179703
0.453931
0.384979
0.3514
0.299894
0.562364
0.288019
0.267072
0.292004
0.602323
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Row 1: 1: 0.828441 7: -0.208968 8: -0.209694 23: -0.40978 43: -0.0340869 44: -1.2837e-16 45: -0.0342097 46: -1.06685e-16 47: -0.069777 48: -2.40693e-16 79: 0.0689149 80: 2.1684e-17 83: 0.0691587 84: -1.9082e-17 233: -1.9082e-17 235: -1.73472e-17
Row 2: 2: 1 11: -0.5 12: -0.5 49: -0.0833333 50: -3.19189e-16 51: -0.0833333 52: -2.70617e-16 99: 0.166667 100: 4.16334e-17 242: -4.85723e-17
Row 3: 3: 0.870927 15: -0.344377 16: -0.336878 30: -0.189673 53: -0.0173948 54: -1.80411e-16 55: -0.0142173 56: -1.31188e-16 57: -0.113542 58: -1.99493e-16 123: 0.074791 124: 7.63278e-17 127: 0.0703636 128: 3.81639e-17 248: -3.46945e-17 250: -8.67362e-18
Row 4: 0: -0.5 4: 1.88347 5: -0.397227 19: -0.177276 20: -0.808965 39: 0 40: 1.76942e-16 41: 0.0833333 42: 9.71445e-17 59: -0.0535736 60: -1.95156e-16 61: -0.164587 62: -2.55872e-16 63: -0.0957505 64: -4.17635e-16 67: 0.119778 68: -1.9082e-17 141: 0.1108 142: -3.81639e-17 227: 0 228: -3.1225e-17 256: -1.73472e-17
Row 5: 4: -0.397227 5: 1.77564 6: -0.326749 20: -0.345183 21: -0.706484 59: -0.028366 60: 1.66317e-16 63: 0.0945705 64: 2.42861e-17 65: -0.0526905 66: -1.65666e-16 67: -0.131261 68: -3.0401e-16 69: -0.0836226 70: -3.77736e-16 73: 0.107149 74: -2.94903e-17 145: 0.0942211 146: -1.9082e-17 228: -1.9082e-17 229: -3.1225e-17 230: -2.08167e-17
Row 6: 5: -0.326749 6: 1.7517 7: -0.273913 21: -0.432506 22: -0.718531 65: -0.0380314 66: 1.5439e-16 69: 0.0924896 70: -3.46945e-18 71: -0.0573097 72: -1.62197e-16 73: -0.116904 74: -3.25261e-16 75: -0.0797051 76: -3.71665e-16 77: 0.102962 78: -4.51028e-17 151: 0.0964985 152: -1.73472e-17 229: -2.77556e-17 231: -3.46945e-17 232: -1.04083e-17
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Row 9: 8: -0.350939 9: 1.74429 10: -0.262838 24: -0.64648 25: -0.484029 81: -0.0557926 82: 1.83881e-16 85: 0.114282 86: -2.77556e-17 87: -0.0342301 88: -1.29237e-16 89: -0.104931 90: -4.15466e-16 91: -0.0957605 92: -3.01842e-16 95: 0.0780365 96: -1.04083e-17 165: 0.0983955 166: 3.46945e-17 236: -3.81639e-17 238: -2.08167e-17 239: -3.81639e-17
Row 10: 9: -0.262838 10: 1.77715 11: -0.267845 25: -0.84644 26: -0.400025 87: -0.0697744 88: 1.65666e-16 91: 0.113581 92: -6.93889e-17 93: -0.0327463 94: -1.26201e-16 95: -0.0777309 96: -4.66641e-16 97: -0.11594 98: -2.83627e-16 101: 0.0773871 102: -6.93889e-18 171: 0.105223 172: 7.80626e-17 238: -4.77049e-17 240: -2.08167e-17 241: -3.46945e-17
Row 11: 2: -0.5 10: -0.267845 11: 1.9829 12: -0.115438 26: -1.09961 49: -6.93889e-17 50: 2.28983e-16 51: 0.0833333 52: 8.67362e-17 93: -0.0708467 94: 1.65666e-16 97: 0.115488 98: -7.28584e-17 99: -0.195755 100: -3.04444e-16 101: -0.0638806 102: -5.75928e-16 105: 0.131662 106: -1.63064e-16 240: -4.42354e-17 242: 5.20417e-17 257: -6.93889e-17
Row 12: 2: -0.5 11: -0.115438 12: 1.81266 13: -0.253469 26: -0.239874 27: -0.703881 49: 0.0833333 50: 9.36751e-17 51: 3.46945e-18 52: 1.8323e-16 99: -0.11965 100: 1.70003e-16 101: 0.0555561 102: -3.81639e-17 103: -0.0432149 104: -1.31405e-16 105: -0.0933383 106: -2.13371e-16 107: -0.0459076 108: -3.18972e-16 111: 0.0854598 112: -2.25514e-17 177: 0.0777613 178: -7.28584e-17 242: -3.46945e-18 243: -2.42861e-17 257: -2.94903e-17 258: -6.93889e-18
Row 13: 12: -0.253469 13: 1.79007 14: -0.404897 27: -0.775929 28: -0.355771 103: -0.0598976 104: 1.44849e-16 107: 0.102142 108: -5.55112e-17 109: -0.0406718 110: -1.83447e-16 111: -0.0608683 112: -4.22405e-16 113: -0.136907 114: -3.06179e-16 117: 0.108155 118: 3.46945e-18 181: 0.0880473 182: 9.19403e-17 243: -3.81639e-17 244: -2.77556e-17 245: -4.16334e-17
Row 14: 13: -0.404897 14: 1.83152 15: -0.218833 28: -0.309609 29: -0.898185 109: -0.0388314 110: 1.82363e-16 113: 0.106314 114: 1.04083e-17 115: -0.0683139 116: -1.45717e-16 117: -0.148866 118: -3.0748e-16 119: -0.0492423 120: -4.21104e-16 121: 0.104786 122: -7.63278e-17 185: 0.0941536 186: -7.28584e-17 244: -2.94903e-17 246: -4.16334e-17 247: -3.46945e-18
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Row 17: 16: -0.20561 17: 1.94288 18: -0.368349 31: -1.16439 32: -0.204537 125: -0.0846663 126: 1.64799e-16 129: 0.118935 130: -1.00614e-16 131: -0.0154075 132: -1.44416e-16 133: -0.0529503 134: -6.01949e-16 135: -0.17079 136: -2.74954e-16 139: 0.076799 140: 3.81639e-17 201: 0.12808 202: 1.63064e-16 251: -6.245e-17 253: -1.04083e-17 254: -5.89806e-17
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Row 20: 4: -0.808965 5: -0.345183 19: -0.545937 20: 3.54246 21: -0.647227 32: -0.584699 34: -0.610448 59: 0.0819397 60: 2.94903e-17 61: 0.122262 62: 1.73472e-17 63: -0.0693744 64: 4.08527e-16 67: -0.10714 68: 3.34802e-16 69: 0.082731 70: -1.9082e-17 141: -0.146695 142: 3.72098e-16 143: 0.115423 144: -5.55112e-17 145: -0.077413 146: -3.60822e-16 147: -0.103474 148: -3.81639e-16 149: -0.0863131 150: -3.36103e-16 155: 0.102553 156: -2.25514e-17 205: 0.0855011 206: -2.60209e-18 228: 4.85723e-17 230: 4.68375e-17 256: -2.25514e-17 259: -4.85723e-17 261: -3.1225e-17 272: -2.08167e-17
Row 21: 5: -0.706484 6: -0.432506 20: -0.647227 21: 3.50609 22: -0.571505 33: -0.714154 34: -0.43421 65: 0.0907219 66: 1.12757e-17 67: 0.118623 68: -1.73472e-17 69: -0.091598 70: 4.00287e-16 73: -0.102619 74: 3.69062e-16 75: 0.0839811 76: -1.56125e-17 145: -0.107223 146: 3.77302e-16 149: 0.0964712 150: -1.04083e-17 151: -0.0880292 152: -3.35235e-16 153: -0.0755245 154: -3.72098e-16 155: -0.119354 156: -3.36536e-16 161: 0.0992989 162: -3.98986e-17 209: 0.0952513 210: 2.42861e-17 229: 6.07153e-17 230: -3.1225e-17 232: 5.72459e-17 260: -2.77556e-17 261: -3.1225e-17 262: -1.73472e-17
Row 22: 6: -0.718531 7: -0.374026 21: -0.571505 22: 3.51428 23: -0.708085 24: -0.536887 33: -0.60525 71: 0.099346 72: 1.38778e-17 73: 0.112374 74: -1.38778e-17 75: -0.0919644 76: 4.06793e-16 77: -0.111917 78: 3.62124e-16 79: 0.0749089 80: -3.20924e-17 151: -0.107744 152: 3.38271e-16 153: 0.0906212 154: -2.77556e-17 157: -0.0555055 158: -3.55185e-16 159: -0.118983 160: -3.39138e-16 161: -0.0995997 162: -3.79904e-16 163: 0.0986109 164: 8.67362e-18 167: 0.109853 168: -3.46945e-17 231: 5.89806e-17 232: -3.1225e-17 234: 4.16334e-17 260: -3.55618e-17 265: -1.73472e-17 266: -2.77556e-17
Row 23: 1: -0.40978 7: -0.968685 8: -1.05698 22: -0.708085 23: 3.76596 24: -0.62243 43: 0.117629 44: -1.00614e-16 45: 0.117335 46: -5.0307e-17 47: -0.166667 48: 3.93782e-16 77: 0.132513 78: -6.93889e-18 79: -0.0886944 80: 4.76615e-16 83: -0.0901615 84: 6.31439e-16 85: 0.148989 86: 4.16334e-17 157: -0.125692 158: 4.18936e-16 159: 0.111193 160: -1.38778e-17 163: -0.156444 164: -5.10009e-16 233: -6.33174e-17 234: -3.1225e-17 235: 6.59195e-17 237: 9.71445e-17 265: -3.46945e-17
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Row 25: 9: -0.484029 10: -0.84644 24: -0.459506 25: 3.57375 26: -0.517151 35: -0.379385 37: -0.887236 87: 0.104004 88: -3.1225e-17 89: 0.0924966 90: -2.08167e-17 91: -0.11583 92: 3.6169e-16 95: -0.0772401 96: 5.20417e-16 97: 0.114309 98: 2.86229e-17 165: -0.124075 166: 3.1225e-16 169: 0.108163 170: -9.54098e-17 171: -0.111385 172: -3.65593e-16 173: -0.113421 174: -2.83627e-16 175: -0.0536744 176: -4.7011e-16 179: 0.0832674 180: -2.42861e-17 219: 0.0933844 220: -9.02056e-17 238: 6.41848e-17 239: -2.77556e-17 241: 6.59195e-17 263: -2.77556e-17 274: -4.85723e-17 276: -4.85723e-17
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Row 27: 12: -0.703881 13: -0.775929 26: -0.788273 27: 3.68527 28: -0.733256 35: -0.683932 103: 0.103112 104: -1.56125e-17 105: 0.159913 106: -6.245e-17 107: -0.145712 108: 4.7011e-16 111: -0.107402 112: 5.29091e-16 113: 0.133611 114: 1.21431e-17 177: -0.124052 178: 4.66641e-16 179: 0.0955182 180: -3.1225e-17 181: -0.133459 182: -4.70977e-16 183: -0.103586 184: -4.21538e-16 187: 0.122057 188: -4.33681e-17 243: 6.41848e-17 245: 8.67362e-17 258: -4.68375e-17 264: -3.29597e-17 270: -3.81639e-17
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Row 39: 0: -0.0833333 4: 0 19: 0.0833333 39: 0.0416667 40: 0.00416667 41: -1.04083e-17 42: 0.00416667 61: -0.0208333 62: 0.00833333 227: -0.00833333
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Row 182: 13: 9.19403e-17 27: -4.70977e-16 28: 3.34151e-16 35: 4.25007e-17 111: 0.00440236 112: 0.000620779 113: -0.00440236 114: -0.00231413 181: 0.00441902 182: 0.0241234 183: 0.0045245 184: -0.000881837 187: -0.0045245 188: 0.0021345 245: -0.00169335 270: -0.00125266
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Row 194: 15: -1.04083e-17 29: -4.92661e-16 30: 4.54281e-16 36: 4.59702e-17 121: 0.004042 122: 0.00171018 123: -0.004042 124: -0.000984488 193: 0.00101989 194: 0.0261493 195: 0.00326207 196: -0.00038455 199: -0.00326207 200: 0.00179017 249: 0.000725692 267: -0.00140562
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Row 196: 28: 2.77556e-17 29: -4.84855e-16 30: 9.02056e-17 36: 3.62557e-16 185: 0.00481769 186: 0.000910568 189: -0.00481769 190: -0.00230122 193: 0.00542411 194: -0.00038455 195: 0.00635644 196: 0.0242762 199: -0.00542411 200: -0.00323152 267: -0.00284697 269: -0.00139066
Row 197: 16: 0.0904872 30: -0.0687924 31: -0.100344 36: 0.078649 127: -0.00659526 128: -0.00422181 129: -0.0160265 130: -0.00610807 197: 0.0777703 198: -0.00157757 199: -0.0184907 200: 0.00789262 203: -0.00117156 204: 0.00442879 252: -0.00580553 268: -0.00838896
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Row 274: 24: -2.77556e-17 25: -4.85723e-17 37: 7.63278e-17 165: -0.00305291 166: 0.00117807 169: -0.00422915 170: 0.00157015 175: -0.00776335 176: 0.000392082 274: 0.0100303
Row 275: 24: -1.38778e-17 33: 6.76542e-17 37: -5.0307e-17 167: -0.00233713 168: 0.00101222 169: -0.00493157 170: 0.00187704 211: -0.00796824 212: 0.000864816 275: 0.010158
Row 276: 25: -4.85723e-17 35: -3.38271e-17 37: 8.32667e-17 173: -0.00231453 174: -0.000442792 175: -0.00702391 176: 0.001127 219: -0.00569553 220: 0.00156979 276: 0.0100226
Row 277: 33: -3.1225e-17 34: 6.93889e-17 38: -3.20924e-17 209: -0.00518112 210: 0.000421735 213: -0.00401931 214: 3.44666e-05 217: -0.00528452 218: -0.000387269 277: 0.00965664
Row 278: 34: -6.93889e-18 36: 5.37764e-17 38: -4.51028e-17 215: -0.00206341 216: -0.000224942 217: -0.00686563 218: 0.0013758 223: -0.0061908 224: 0.00160074 278: 0.0100799
Row 279: 33: -3.46945e-18 37: -4.33681e-17 38: 5.11743e-17 211: -0.00749901 212: 0.0013602 213: -0.00185554 214: -0.000520962 225: -0.00593612 226: -0.00188116 279: 0.0101938
Row 280: 28: -2.77556e-17 35: 7.45931e-17 38: -4.64039e-17 187: -0.0048394 188: 0.00147685 191: -0.00283327 192: 0.000808145 221: -0.00726384 222: -0.00066871 280: 0.00995768
Row 281: 28: -2.42861e-17 36: -4.68375e-17 38: 6.76542e-17 189: -0.00450931 190: 0.00069548 191: -0.00397987 192: 0.000519 223: -0.00606631 224: -0.00017648 281: 0.00970366
Row 282: 35: -2.77556e-17 37: 5.89806e-17 38: -3.38271e-17 219: -0.00555242 220: 0.000252739 221: -0.00408933 222: -0.00023496 225: -0.00484754 226: -0.000487698 282: 0.00965953
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]:
[8]:
Draw (-grad(gfu), mesh, "Flux", vectors= { "grid_size" : 40})
[8]:
Calculate error:
[9]:
exact = 16*x*(1-x)*y*(1-y)
print ("L2-error:", sqrt(Integrate((gfu-exact)**2, mesh)))
L2-error: 5.2604701627741786e-05
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