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:

In [1]:
import netgen.gui
%gui tk
from ngsolve import *
from netgen.geom2d import unit_square

2. Generate an unstructured mesh

In [2]:
mesh = Mesh(unit_square.GenerateMesh(maxh=0.2))
mesh.nv, mesh.ne   # number of vertices & elements
Out[2]:
(37, 52)

  • Here we prescribed a maximal mesh-size of 0.2 using the maxh flag.

  • The mesh can be viewed by switching to the Mesh tab in the Netgen GUI.

3. Declare a finite element space:

In [3]:
fes = H1(mesh, order=2, dirichlet="bottom|right")
fes.ndof  # number of unknowns in this space
Out[3]:
125

Python's help system displays further documentation.

In [4]:
help(fes)

Help on FESpace in module ngsolve.comp object:

class FESpace(pybind11_builtins.pybind11_object_56)
 |  Finite Element Space
 |  
 |  Provides the functionality for finite element calculations. Use
 |  the finite element space generator functions to construct the space,
 |  this can sometimes provide additional functionality (e.g HCurl).
 |  When createing a FESpace with a generator function the set parameters
 |  are passed to the FESpace constructor with and the type parameter is
 |  set. If the space has additional functionality it is added to the space.
 |  
 |  Available generator functions
 |  
 |  H1
 |  HCurl
 |  HDiv
 |  L2
 |  FacetFESpace
 |  HDivDiv
 |  
 |  2 __init__ overloads:
 |    1) To create a registered FESpace
 |    2) To create a compound FESpace from multiple created FESpaces
 |  
 |  1)
 |  
 |  Parameters
 |  
 |  type : string
 |    Type of the finite element space. This parameter is automatically
 |    set if the space is constructed with a generator function.
 |  
 |  mesh : ngsolve.Mesh
 |    Mesh on which the finite element space is defined on.
 |  
 |  flags : dict
 |    Provide additional flags for the finite element space, possible options
 |    are:
 |      dgjumps : bool
 |        Enable DG functionality
 |      print : bool
 |        Write additional debug information to testout file. This
 |        file must be set by ngsolve.SetTestoutFile.
 |  
 |  order : int
 |    Order of the finite element space
 |  
 |  is_complex : bool
 |    Set to true if you want to specify complex (bi-)linearforms on the
 |    FESpace.
 |  
 |  dirichlet : regexpr
 |    Regular expression string defining the dirichlet boundary.
 |    More than one boundary can be combined by the | operator,
 |    Example: dirichlet = "dirichlet1|dirichlet2"
 |  
 |  definedon : list of bits
 |    Define FESpace only on the given domain numbers. Must be list of
 |    0s and 1s, 0 for not defined, 1 for defined.
 |  
 |  dim : int
 |    Create multi dimensional FESpace (i.e. [H1]^3)
 |  
 |  2)
 |  
 |  Parameters:
 |  
 |  spaces : list of ngsolve.FESpace
 |    List of the spaces for the compound finite element space
 |  
 |  flags : dict
 |      Additional flags for the compound FESpace
 |  
 |  Method resolution order:
 |      FESpace
 |      pybind11_builtins.pybind11_object_56
 |      builtins.object
 |  
 |  Methods defined here:
 |  
 |  CouplingType(...)
 |      CouplingType(self: ngsolve.comp.FESpace, dofnr: int) -> ngsolve.comp.COUPLING_TYPE
 |      
 |      get coupling type of a degree of freedom
 |  
 |  Elements(...)
 |      Elements(*args, **kwargs)
 |      Overloaded function.
 |      
 |      1. Elements(self: ngsolve.comp.FESpace, VOL_or_BND: ngsolve.comp.VorB=VorB.VOL, heapsize: int=10000) -> ngsolve.comp.FESpaceElementRange
 |      
 |      2. Elements(self: ngsolve.comp.FESpace, VOL_or_BND: ngsolve.comp.VorB=VorB.VOL, heap: ngsolve.ngstd.LocalHeap) -> ngsolve.comp.FESpaceElementRange
 |  
 |  FreeDofs(...)
 |      FreeDofs(self: ngsolve.comp.FESpace, coupling: bool=False) -> ngsolve.ngstd.BitArray
 |  
 |  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
 |  
 |  GetFE(...)
 |      GetFE(*args, **kwargs)
 |      Overloaded function.
 |      
 |      1. GetFE(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.ElementId) -> object
 |      
 |      2. GetFE(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.ElementId, arg1: ngsolve.ngstd.LocalHeap) -> ngsolve.fem.FiniteElement
 |  
 |  Range(...)
 |      Range(self: ngsolve.comp.FESpace, arg0: int) -> slice
 |  
 |  SetCouplingType(...)
 |      SetCouplingType(self: ngsolve.comp.FESpace, dofnr: int, coupling_type: ngsolve.comp.COUPLING_TYPE) -> None
 |      
 |      set coupling type of a degree of freedom
 |  
 |  SetOrder(...)
 |      SetOrder(*args, **kwargs)
 |      Overloaded function.
 |      
 |      1. SetOrder(self: ngsolve.comp.FESpace, element_type: ngsolve.fem.ET, order: object=<ngsolve.ngstd.DummyArgument>, order_left: object=<ngsolve.ngstd.DummyArgument>, order_right: object=<ngsolve.ngstd.DummyArgument>) -> None
 |      
 |      2. SetOrder(self: ngsolve.comp.FESpace, nodeid: ngsolve.comp.NodeId, order: int) -> None
 |  
 |  SolveM(...)
 |      SolveM(self: ngsolve.comp.FESpace, rho: ngsolve.fem.CoefficientFunction, vec: ngsolve.la.BaseVector, heapsize: int=1000000) -> None
 |  
 |  TestFunction(...)
 |      TestFunction(self: ngsolve.comp.FESpace) -> object
 |      
 |      Gives a proxy to be used as a testfunction for Symbolic Integrators
 |  
 |  TrialFunction(...)
 |      TrialFunction(self: ngsolve.comp.FESpace) -> object
 |      
 |      Gives a proxy to be used as a trialfunction in Symbolic Integrators
 |  
 |  Update(...)
 |      Update(self: ngsolve.comp.FESpace, heapsize: int=1000000) -> None
 |      
 |      update space after mesh-refinement
 |  
 |  __eq__(...)
 |      __eq__(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace) -> bool
 |  
 |  __init__(self, /, *args, **kwargs)
 |      Initialize self.  See help(type(self)) for accurate signature.
 |  
 |  __new__ = patched_new(class_t, *args, **kwargs)
 |  
 |  __ngsid__(...)
 |      __ngsid__(self: ngsolve.comp.FESpace) -> int
 |  
 |  __reduce__(...)
 |      __reduce__(self: object) -> tuple
 |  
 |  __setstate__(...)
 |      __setstate__(self: object, arg0: tuple) -> None
 |  
 |  __str__(...)
 |      __str__(self: ngsolve.comp.FESpace) -> str
 |  
 |  __timing__(...)
 |      __timing__(self: ngsolve.comp.FESpace) -> object
 |  
 |  ----------------------------------------------------------------------
 |  Data descriptors defined here:
 |  
 |  __dict__
 |  
 |  components
 |      list of gridfunctions for compound gridfunction
 |  
 |  globalorder
 |      query global order of space
 |  
 |  mesh
 |  
 |  ndof
 |      number of degrees of freedom
 |  
 |  ndofglobal
 |      global number of dofs on MPI-distributed mesh
 |  
 |  order
 |      proxy to set order for individual nodes
 |  
 |  type
 |      type of finite element space

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.

In [5]:
u = fes.TrialFunction()  # symbolic object
v = fes.TestFunction()   # symbolic object
gfu = GridFunction(fes)  # solution

5. Define and assemble linear and bilinear forms:

In [6]:
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:

In [7]:
print(f.vec)

 0.000333333
 0.0103012
 0.00633333
 0.000333333
 0.00387671
 0.006844
 0.00946082
 0.0131276
 0.00932238
 0.0231198
 0.0160023
 0.024313
 0.0168221
 0.0111949
 0.00812513
 0.00410081
 0.00276968
 0.000493909
 0.00137444
 0.00297784
 0.0117697
 0.0200805
 0.0283719
 0.0282328
 0.0279711
 0.0134645
 0.00752298
 0.0155354
 0.0257406
 0.0228613
 0.0214477
 0.00993661
 0.0137531
 0.0156134
 0.0216362
 0.0237827
 0.0210528
 -6.66667e-05
 -3.33333e-05
 -0.000687449
 -0.000587035
 -0.00126541
 -0.0008
 -0.000766667
 -6.66667e-05
 -3.33333e-05
 -0.000228362
 -0.000220805
 -0.00042906
 -0.00033644
 -0.000514805
 -0.000625037
 -0.000462964
 -0.000842943
 -0.000719098
 -0.000956807
 -0.00116943
 -0.000587035
 -0.00113898
 -0.000776995
 -0.00136243
 -0.00146798
 -0.00149124
 -0.000668081
 -0.00130141
 -0.00120446
 -0.00164251
 -0.00152774
 -0.00132933
 -0.00049277
 -0.00133031
 -0.000399239
 -0.00100556
 -0.000902792
 -0.000250598
 -0.00060796
 -0.000769314
 -0.000227209
 -0.000456957
 -2.46955e-05
 -0.000350467
 -0.0001952
 -2.46955e-05
 -9.87818e-05
 -3.89193e-05
 -0.000155955
 -0.000192762
 -0.000355936
 -0.000244359
 -0.000594642
 -0.000462583
 -0.000560307
 -0.000951433
 -0.000705779
 -0.000849162
 -0.00071472
 -0.00122166
 -0.00121405
 -0.00104716
 -0.0010076
 -0.00102318
 -0.00114876
 -0.000998108
 -0.000993348
 -0.000471213
 -0.000771239
 -0.000693185
 -0.000388595
 -0.000563781
 -0.000620059
 -0.00061514
 -0.000777933
 -0.000893067
 -0.000986579
 -0.000911771
 -0.00100144
 -0.000578666
 -0.000611576
 -0.000615209
 -0.000710176
 -0.000769118
 -0.000760015
 -0.00096259
 -0.000889339
 -0.000981825
In [8]:
print(a.mat)

Row 0:   0: 1
Row 1:   1: 0.870166
Row 2:   2: 1
Row 3:   3: 1
Row 4:   0: -0.5   4: 1.89594
Row 5:   4: -0.328601   5: 1.75373
Row 6:   5: -0.252529   6: 1.74397
Row 7:   1: -0.355392   6: -0.246158   7: 1.75039
Row 8:   1: -0.315451   8: 2.11144
Row 9:   8: -0.417618   9: 1.74594
Row 10:   9: -0.35985   10: 1.7563
Row 11:   2: -0.5   10: -0.267891   11: 1.8234
Row 12:   2: -0.5   11: -0.157475   12: 1.9414
Row 13:   12: -0.279511   13: 1.73971
Row 14:   13: -0.347754   14: 1.75594
Row 15:   3: -0.5   14: -0.389763   15: 1.87732
Row 16:   3: -0.5   15: -0.203686   16: 1.80816
Row 17:   16: -0.436369   17: 2.02265
Row 18:   17: -0.424487   18: 1.88532
Row 19:   0: -0.5   4: -0.19922   18: -0.505055   19: 1.8246
Row 20:   4: -0.868115   5: -0.442585   19: -0.376723   20: 3.59693
Row 21:   6: -0.713275   7: -0.495693   21: 3.52299
Row 22:   1: -0.199323   7: -0.653151   8: -1.37837   9: -0.102645   21: -0.73776   22: 3.94083
Row 23:   9: -0.484277   10: -0.45587   23: 3.57987
Row 24:   11: -0.1789   12: -1.00442   13: -0.54338   24: 3.70855
Row 25:   14: -0.437784   15: -0.783873   16: -0.40993   25: 3.54731
Row 26:   16: -0.258171   17: -1.1618   18: -0.149049   25: -0.770254   26: 3.78149
Row 27:   5: -0.730015   6: -0.532007   20: -0.57912   21: -0.447455   27: 3.51184
Row 28:   9: -0.381555   21: -0.393059   22: -0.869584   23: -0.884276   28: 3.62025
Row 29:   10: -0.672687   11: -0.719133   23: -0.86098   24: -0.905134   29: 3.7132
Row 30:   13: -0.569066   14: -0.580637   24: -0.482194   25: -0.65918   30: 3.49528
Row 31:   18: -0.806729   19: -0.243604   20: -0.822289   26: -0.783458   31: 3.67336
Row 32:   21: -0.735752   27: -0.733473   28: -0.670829   32: 3.63948
Row 33:   20: -0.508096   27: -0.489774   31: -0.764287   32: -0.695649   33: 3.53766
Row 34:   25: -0.486293   26: -0.658754   30: -0.483077   31: -0.252992   33: -0.601314   34: 3.41839
Row 35:   23: -0.419347   24: -0.594525   29: -0.555266   30: -0.721121   34: -0.401719   35: 3.54681
Row 36:   23: -0.475121   28: -0.420952   32: -0.80378   33: -0.478544   34: -0.534244   35: -0.854834   36: 3.56748
Row 37:   0: -0.0833333   4: 0   19: 0.0833333   37: 0.0416667
Row 38:   0: -0.0833333   4: 0.0833333   19: -3.00685e-18   37: 1.73472e-18   38: 0.0416667
Row 39:   1: -0.0247066   7: -0.065298   22: 0.0900047   39: 0.0373091
Row 40:   1: -0.00851394   8: -0.10635   22: 0.114864   40: 0.0418598
Row 41:   1: -0.111807   7: 0.12453   8: 0.158925   22: -0.171648   39: -0.0163245   40: -0.0265875   41: 0.0791689
Row 42:   2: -0.0833333   11: 2.65991e-18   12: 0.0833333   42: 0.0416667
Row 43:   2: -0.0833333   11: 0.0833333   12: 0   42: 0   43: 0.0416667
Row 44:   3: -0.0833333   15: 2.65991e-18   16: 0.0833333   44: 0.0416667
Row 45:   3: -0.0833333   15: 0.0833333   16: 0   44: 0   45: 0.0416667
Row 46:   4: -0.0517721   5: -0.0385685   20: 0.0903406   46: 0.0362769
Row 47:   0: 0.166667   4: -0.176247   19: -0.113935   20: 0.123515   37: -0.0208333   38: -0.0208333   47: 0.0808464
Row 48:   4: -0.0879702   5: 0.0933354   19: 0.063805   20: -0.0691702   46: -0.00964212   47: -0.00765043   48: 0.0754566
Row 49:   5: -0.0563741   6: -0.0464315   27: 0.102806   49: 0.0362235
Row 50:   4: 0.106539   5: -0.120062   20: -0.0980085   27: 0.111532   46: -0.012943   48: -0.0136917   50: 0.0729587
Row 51:   5: -0.0772839   6: 0.0885198   20: 0.0814321   27: -0.092668   49: -0.0116079   50: -0.0115591   51: 0.0729053
Row 52:   6: -0.0608303   7: -0.0436771   21: 0.104507   52: 0.0363834
Row 53:   6: -0.0832626   7: 0.0847034   21: -0.088476   27: 0.0870353   52: -0.0109193   53: 0.0726545
Row 54:   5: 0.0984623   6: -0.100137   21: 0.102848   27: -0.101173   49: -0.0140935   51: -0.0105221   53: -0.0111997   54: 0.0724945
Row 55:   6: 0.101857   7: -0.0845868   21: -0.124446   22: 0.107177   52: -0.0152076   53: -0.0102566   55: 0.0729122
Row 56:   1: 0.0839386   7: -0.0981704   21: 0.102555   22: -0.0883228   39: -0.00617666   41: -0.014808   55: -0.015904   56: 0.0738379
Row 57:   8: -0.123378   9: 0.00851394   22: 0.114864   57: 0.0461167
Row 58:   1: 0.0610891   8: -0.122178   9: 0.0610891   22: 3.46945e-18   40: -0.00212848   41: -0.0131438   57: 0.00212848   58: 0.0879765
Row 59:   9: -0.0356793   10: -0.0502285   23: 0.0859078   59: 0.0364707
Row 60:   8: 0.192981   9: -0.107927   22: -0.216622   28: 0.131568   57: -0.0308445   58: -0.0174008   60: 0.0854142
Row 61:   9: -0.0852437   10: 0.110204   23: -0.132822   28: 0.107862   59: -0.0125571   61: 0.0746947
Row 62:   9: -0.0706549   22: 0.118866   23: 0.127627   28: -0.175838   60: -0.0233112   61: -0.0206484   62: 0.0775214
Row 63:   10: -0.0584739   11: -0.0420246   29: 0.100499   63: 0.0362867
Row 64:   9: 0.0956543   10: -0.113616   23: -0.105753   29: 0.123715   59: -0.00891982   61: -0.0149938   64: 0.0738369
Row 65:   10: -0.0703982   11: 0.086673   23: 0.0958239   29: -0.112099   63: -0.0105061   64: -0.0175186   65: 0.0736529
Row 66:   2: 0.166667   11: -0.115053   12: -0.188774   24: 0.137161   42: -0.0208333   43: -0.0208333   66: 0.0825183
Row 67:   11: -0.104077   12: 0.131687   24: -0.198853   29: 0.171242   66: -0.0263603   67: 0.0831865
Row 68:   10: 0.103122   11: -0.0427455   24: 0.0915086   29: -0.151886   63: -0.0146185   65: -0.0111621   67: -0.0233529   68: 0.0786216
Row 69:   12: -0.0619621   13: -0.037384   24: 0.0993461   69: 0.0364828
Row 70:   11: 0.0579654   12: -0.0728309   13: 0.0839692   24: -0.0691036   66: -0.00792991   67: -0.00656144   69: -0.009346   70: 0.0773344
Row 71:   13: -0.0456943   14: -0.0414464   30: 0.0871407   71: 0.0362749
Row 72:   12: 0.108547   13: -0.0957353   24: -0.104283   30: 0.091471   69: -0.0154905   70: -0.0116463   72: 0.0726454
Row 73:   13: -0.111138   14: 0.0994054   24: 0.0955002   30: -0.0837673   71: -0.0103616   72: -0.0105802   73: 0.0724375
Row 74:   14: -0.0328517   15: -0.0491798   25: 0.0820314   74: 0.036748
Row 75:   14: -0.120287   15: 0.11414   25: -0.0986898   30: 0.104836   74: -0.0122949   75: 0.0729852
Row 76:   13: 0.103653   14: -0.0980712   25: 0.0896223   30: -0.0952042   71: -0.0114236   73: -0.0144897   75: -0.0123775   76: 0.0725121
Row 77:   3: 0.166667   15: -0.164799   16: -0.119541   25: 0.117674   44: -0.0208333   45: -0.0208333   77: 0.079572
Row 78:   14: 0.0978122   15: -0.0989082   16: 0.0701555   25: -0.0690595   74: -0.00821292   75: -0.0162401   77: -0.00905195   78: 0.0746533
Row 79:   16: 0.000990177   17: -0.0978067   26: 0.0968166   79: 0.0423862
Row 80:   15: 0.115413   16: -0.0779664   25: -0.154113   26: 0.116666   77: -0.0203664   78: -0.00848692   80: 0.0751003
Row 81:   16: -0.104842   17: 0.170535   25: 0.104761   26: -0.170454   79: -0.0244517   80: -0.0181619   81: 0.0795812
Row 82:   17: -0.0958264   18: -0.000990177   26: 0.0968166   82: 0.0418911
Row 83:   16: 0.071738   17: -0.143476   18: 0.071738   26: -4.98733e-18   79: 0.000247544   81: -0.018182   82: -0.000247544   83: 0.0842773
Row 84:   18: -0.0707206   19: -0.00640091   31: 0.0771215   84: 0.0403243
Row 85:   17: 0.166574   18: -0.134482   26: -0.157758   31: 0.125666   82: -0.0239566   83: -0.017687   85: 0.0792703
Row 86:   18: -0.108027   19: 0.0905768   26: 0.0857827   31: -0.0683322   84: -0.00160023   85: -0.0154828   86: 0.0777036
Row 87:   4: 0.126117   19: -0.067403   20: -0.180941   31: 0.122227   47: -0.0232284   48: -0.00830083   87: 0.0777828
Row 88:   18: 0.154896   19: -0.116361   20: 0.120213   31: -0.158748   84: -0.0176801   86: -0.021044   87: -0.0220068   88: 0.0789275
Row 89:   5: 0.100491   20: -0.0760107   27: -0.118997   33: 0.0945165   50: -0.0163238   51: -0.00879891   89: 0.0728818
Row 90:   19: 0.0663852   20: -0.076053   31: -0.0858102   33: 0.095478   87: -0.00854993   88: -0.00804637   90: 0.0747279
Row 91:   20: -0.0993044   27: 0.103985   31: 0.100631   33: -0.105312   89: -0.0134253   90: -0.0129026   91: 0.0723247
Row 92:   7: 0.082499   21: -0.0738587   22: -0.0952466   28: 0.0866064   55: -0.0108901   56: -0.00973461   92: 0.0730163
Row 93:   6: 0.100285   21: -0.117804   27: -0.0999266   32: 0.117445   53: -0.0105591   54: -0.0145122   93: 0.0730766
Row 94:   21: -0.122214   22: 0.11103   28: -0.105413   32: 0.116597   92: -0.0129215   94: 0.0732841
Row 95:   21: -0.0603663   27: 0.0874671   28: 0.0843162   32: -0.111417   93: -0.0144226   94: -0.0134317   95: 0.0736022
Row 96:   9: 0.0639451   21: 0.0942641   22: -0.0849651   28: -0.0732441   60: -0.00958094   62: -0.00640534   92: -0.00873008   94: -0.0148359   96: 0.075785
Row 97:   9: 0.0703022   23: -0.0823257   28: -0.0696312   36: 0.0816547   61: -0.00631716   62: -0.0112584   97: 0.0748341
Row 98:   10: 0.0793903   23: -0.0549424   29: -0.100427   35: 0.0759796   64: -0.0134101   65: -0.00643743   98: 0.0747166
Row 99:   23: -0.115317   29: 0.120209   35: -0.110223   36: 0.105331   98: -0.0116967   99: 0.0738579
Row 100:   23: -0.105484   28: 0.109148   35: 0.104135   36: -0.107799   97: -0.0110906   99: -0.0158591   100: 0.0731175
Row 101:   11: 0.075928   24: -0.0399176   29: -0.123588   35: 0.087578   67: -0.0194577   68: 0.000475734   101: 0.0785904
Row 102:   13: 0.102329   24: -0.110446   30: -0.0991518   35: 0.107269   72: -0.0122875   73: -0.0132948   102: 0.0724909
Row 103:   24: -0.0954889   29: 0.103202   30: 0.0880464   35: -0.0957591   101: -0.0114394   102: -0.0125004   103: 0.0725839
Row 104:   16: 0.0761325   25: -0.0704368   26: -0.095148   34: 0.0894523   80: -0.0110047   81: -0.00802846   104: 0.0734901
Row 105:   14: 0.0954388   25: -0.0828381   30: -0.0976803   34: 0.0850796   75: -0.0138316   76: -0.0100281   105: 0.0725954
Row 106:   25: -0.116082   26: 0.106858   30: 0.102707   34: -0.0934831   104: -0.0127823   105: -0.0105885   106: 0.0726534
Row 107:   18: 0.0875857   26: -0.0825145   31: -0.0866513   34: 0.0815801   85: -0.0159336   86: -0.00596284   107: 0.0749355
Row 108:   25: 0.0940512   26: -0.124373   31: 0.091562   34: -0.06124   104: -0.00958075   106: -0.0139321   107: -0.00572925   108: 0.0738514
Row 109:   21: 0.089532   27: -0.0577045   32: -0.11535   33: 0.0835224   93: -0.0149388   95: -0.00744421   109: 0.073825
Row 110:   20: 0.0910986   27: -0.114839   32: 0.12015   33: -0.0964098   89: -0.0102038   91: -0.0125709   109: -0.0138987   110: 0.0732194
Row 111:   21: 0.0934596   28: -0.0563856   32: -0.127804   36: 0.0907298   94: -0.0157175   95: -0.00764736   111: 0.0739985
Row 112:   23: 0.102078   28: -0.122864   32: 0.123012   36: -0.102226   97: -0.00932305   100: -0.0161965   111: -0.0162334   112: 0.0738121
Row 113:   23: 0.102615   24: 0.0992647   29: -0.130867   35: -0.0710132   98: -0.00729817   99: -0.0183556   101: -0.0104551   103: -0.014361   113: 0.0736061
Row 114:   25: 0.103079   30: -0.130539   34: -0.0821025   35: 0.109562   105: -0.0106815   106: -0.0150883   114: 0.0732885
Row 115:   24: 0.0953117   30: -0.0762038   34: 0.0775358   35: -0.0966436   102: -0.0143167   103: -0.00951119   114: -0.00984418   115: 0.0732586
Row 116:   20: 0.0928885   31: -0.0682693   33: -0.101603   34: 0.076984   90: -0.0109669   91: -0.0122552   116: 0.0743134
Row 117:   26: 0.127308   31: -0.144416   33: 0.133506   34: -0.116399   107: -0.0146658   108: -0.0171612   116: -0.0144339   117: 0.0757449
Row 118:   27: 0.092483   32: -0.133585   33: -0.0494054   36: 0.0905075   109: -0.00698191   110: -0.0161388   118: 0.074733
Row 119:   28: 0.0838742   32: -0.118425   33: 0.0818246   36: -0.0472739   111: -0.00644903   112: -0.0145195   118: -0.00536944   119: 0.0749155
Row 120:   31: 0.095019   33: -0.13405   34: -0.0635953   36: 0.102626   116: -0.00481209   117: -0.0189427   120: 0.0744661
Row 121:   32: 0.129377   33: -0.10283   34: 0.0868304   36: -0.113377   118: -0.0172574   119: -0.0150867   120: -0.0110867   121: 0.0739911
Row 122:   30: 0.108344   34: -0.0828528   35: -0.149221   36: 0.12373   114: -0.0175463   115: -0.00953977   122: 0.0747568
Row 123:   33: 0.100763   34: -0.0700598   35: 0.106613   36: -0.137316   120: -0.0145699   121: -0.0106209   122: -0.019759   123: 0.074104
Row 124:   23: 0.0825931   34: 0.0722701   35: -0.0682748   36: -0.0865885   99: -0.0104737   100: -0.0101746   122: -0.0111734   123: -0.0068941   124: 0.0743339

6. Solve the system:

In [9]:
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:

In [10]:
print(gfu.vec)

       0
       0
       0
 0.0924091
       0
       0
       0
       0
       0
       0
       0
       0
 0.0579249
 0.0863395
 0.095412
 0.0945061
 0.0887126
 0.0780983
 0.0595882
 0.0330527
 0.0347201
 0.0306088
 0.0226997
 0.0485255
 0.0702498
 0.0920839
 0.0799272
 0.0334133
 0.0451654
 0.0475731
 0.089964
 0.0608074
 0.0521306
 0.0604622
 0.0816294
 0.0755969
 0.067892
       0
 -0.00580903
       0
       0
 0.0247054
       0
 -0.0349026
 0.00339392
 -0.00739309
       0
 -0.0100181
 -0.00505004
       0
 -0.0147034
 -0.00938597
       0
 -0.017922
 -0.0121489
 -0.00529069
 -0.0292704
       0
 -0.0183813
       0
 -0.0461399
 -0.0215322
 -0.05256
       0
 -0.0378931
 -0.0206654
 -0.0330051
 -0.0606934
 -0.0276814
 -0.024247
 -0.00671107
 -0.0145564
 -0.0107763
 -0.00927653
 -0.00544311
 -0.00971665
 -0.00812781
 -0.00479834
 -0.00711323
 -0.00781592
 0.00251074
 -0.00483204
 -0.00805012
 0.00347202
 -0.0083864
 -0.0049253
 0.00455957
 0.00401028
 -0.00241395
 -0.00346299
 -0.00961795
 -0.00949918
 -0.0143154
 -0.00652048
 -0.0197189
 -0.00467488
 -0.000876779
 -0.00958371
 -0.00431468
 -0.0129541
 -0.0262452
 -0.00799141
 -0.0244375
 -0.00989479
 -0.00826822
 -0.0118755
 -0.00901224
 -0.00914765
 -0.0047194
 -0.0107081
 -0.0105955
 -0.0151661
 -0.00862588
 -0.0250001
 -0.0135096
 -0.00898515
 -0.00363723
 -0.0139331
 -0.00600571
 -0.00733149
 -0.0111797
 -0.0153338
 -0.0227022
 -0.011174
 -0.00783562

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 netgen on the command line:

      netgen poisson.py

    You can then ask for a python shell from the GUI's menu options (Solve -> Python shell).

In [11]: