This page was generated from unit-1.1-poisson/poisson.ipynb.
1.1 First NGSolve example¶
Let us solve the Poisson problem of finding \(u\) satisfying
\[\begin{split}\begin{aligned}
-\Delta u & = f && \text { in the unit square},
\\
u & = 0 && \text{ on the bottom and right parts of the boundary},
\\
\frac{\partial u }{\partial n } & = 0
&& \text{ on the remaining boundary parts}.
\end{aligned}\end{split}\]
Quick steps to solution:¶
1. Import NGSolve and Netgen Python modules:¶
[1]:
from ngsolve import *
from ngsolve.webgui import Draw
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 maxh flag.
[3]:
Draw(mesh);
3. Declare a finite element space:¶
[4]:
fes = H1(mesh, order=2, dirichlet="bottom|right")
fes.ndof # number of unknowns in this space
[4]:
133
Python’s help system displays further documentation.
[5]:
help(fes)
Help on H1 in module ngsolve.comp object:
class H1(FESpace)
| An H1-conforming finite element space.
|
| The H1 finite element space consists of continuous and
| element-wise polynomial functions. It uses a hierarchical (=modal)
| basis built from integrated Legendre polynomials on tensor-product elements,
| and Jaboci polynomials on simplicial elements.
|
| Boundary values are well defined. The function can be used directly on the
| boundary, using the trace operator is optional.
|
| The H1 space supports variable order, which can be set individually for edges,
| faces and cells.
|
| Internal degrees of freedom are declared as local dofs and are eliminated
| if static condensation is on.
|
| The wirebasket consists of all vertex dofs. Optionally, one can include the
| first (the quadratic bubble) edge basis function, or all edge basis functions
| into the wirebasket.
|
| Keyword arguments can be:
|
| order: int = 1
| order of finite element space
| complex: bool = False
| Set if FESpace should be complex
| dirichlet: regexpr
| Regular expression string defining the dirichlet boundary.
| More than one boundary can be combined by the | operator,
| i.e.: dirichlet = 'top|right'
| dirichlet_bbnd: regexpr
| Regular expression string defining the dirichlet bboundary,
| i.e. points in 2D and edges in 3D.
| More than one boundary can be combined by the | operator,
| i.e.: dirichlet_bbnd = 'top|right'
| dirichlet_bbbnd: regexpr
| Regular expression string defining the dirichlet bbboundary,
| i.e. points in 3D.
| More than one boundary can be combined by the | operator,
| i.e.: dirichlet_bbbnd = '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.
| autoupdate: bool = False
| Automatically update on a change to the mesh.
| low_order_space: bool = True
| Generate a lowest order space together with the high-order space,
| needed for some preconditioners.
| hoprolongation: bool = False
| Create high order prolongation operators,
| only available for H1 and L2 on simplicial meshes
| order_policy: ORDER_POLICY = ORDER_POLICY.OLDSTYLE
| CONSTANT .. use the same fixed order for all elements,
| NODAL ..... use the same order for nodes of same shape,
| VARIABLE ... use an individual order for each edge, face and cell,
| OLDSTYLE .. as it used to be for the last decade
| print: bool = False
| (historic) print some output into file set by 'SetTestoutFile'
| wb_withedges: bool = true(3D) / false(2D)
| use lowest-order edge dofs for BDDC wirebasket
| wb_fulledges: bool = false
| use all edge dofs for BDDC wirebasket
| hoprolongation: bool = false
| (experimental, only trigs) creates high order prolongation,
| and switches off low-order space
|
| Method resolution order:
| H1
| FESpace
| NGS_Object
| 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
|
| ----------------------------------------------------------------------
| Static methods defined here:
|
| __flags_doc__(...) from builtins.PyCapsule
| __flags_doc__() -> dict
|
| ----------------------------------------------------------------------
| Data descriptors defined here:
|
| __dict__
|
| ----------------------------------------------------------------------
| Methods inherited from FESpace:
|
| ApplyM(...)
| ApplyM(self: ngsolve.comp.FESpace, vec: ngsolve.la.BaseVector, rho: ngsolve.fem.CoefficientFunction = None, definedon: ngsolve.comp.Region = None) -> None
|
| Apply mass-matrix. Available only for L2-like spaces
|
| ConvertL2Operator(...)
| ConvertL2Operator(self: ngsolve.comp.FESpace, l2space: ngsolve.comp.FESpace) -> BaseMatrix
|
| CouplingType(...)
| CouplingType(self: ngsolve.comp.FESpace, dofnr: int) -> ngsolve.comp.COUPLING_TYPE
|
|
| Get coupling type of a degree of freedom.
|
| Parameters:
|
| dofnr : int
| input dof number
|
| CreateDirectSolverCluster(...)
| CreateDirectSolverCluster(self: ngsolve.comp.FESpace, **kwargs) -> list
|
| CreateSmoothingBlocks(...)
| CreateSmoothingBlocks(self: ngsolve.comp.FESpace, **kwargs) -> pyngcore.pyngcore.Table_I
|
|
| Create table of smoothing blocks for block-Jacobi/block-Gauss-Seidel preconditioners.
|
| Every table entry describes the set of dofs belonging a Jacobi/Gauss-Seidel block.
|
| Paramters:
|
| blocktype: string | [ string ] | int
| describes blocktype.
| string form ["vertex", "edge", "face", "facet", "vertexedge", ....]
| or list of strings for combining multiple blocktypes
| int is for backward compatibility with old style blocktypes
|
| condense: bool = False
| boolexclude dofs eliminated by static condensation
|
| Elements(...)
| Elements(*args, **kwargs)
| Overloaded function.
|
| 1. Elements(self: ngsolve.comp.FESpace, VOL_or_BND: ngsolve.comp.VorB = <VorB.VOL: 0>) -> ngsolve.comp.FESpaceElementRange
|
|
| Returns an iterable range of elements.
|
| Parameters:
|
| VOL_or_BND : ngsolve.comp.VorB
| input VOL, BND, BBND,...
|
|
|
| 2. Elements(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.Region) -> Iterator[ngsolve.comp.FESpaceElement]
|
| FinalizeUpdate(...)
| FinalizeUpdate(self: ngsolve.comp.FESpace) -> None
|
| finalize update
|
| FreeDofs(...)
| FreeDofs(self: ngsolve.comp.FESpace, coupling: bool = False) -> pyngcore.pyngcore.BitArray
|
|
|
| Return BitArray of free (non-Dirichlet) dofs\n
| coupling=False ... all free dofs including local dofs\n
| coupling=True .... only element-boundary free dofs
|
| Parameters:
|
| coupling : bool
| input coupling
|
| GetDofNrs(...)
| GetDofNrs(*args, **kwargs)
| Overloaded function.
|
| 1. GetDofNrs(self: ngsolve.comp.FESpace, ei: ngsolve.comp.ElementId) -> tuple
|
|
|
| Parameters:
|
| ei : ngsolve.comp.ElementId
| input element id
|
|
|
| 2. GetDofNrs(self: ngsolve.comp.FESpace, ni: ngsolve.comp.NodeId) -> tuple
|
|
|
| Parameters:
|
| ni : ngsolve.comp.NodeId
| input node id
|
| GetDofs(...)
| GetDofs(self: ngsolve.comp.FESpace, region: ngsolve.comp.Region) -> pyngcore.pyngcore.BitArray
|
|
| Returns all degrees of freedom in given region.
|
| Parameters:
|
| region : ngsolve.comp.Region
| input region
|
| GetFE(...)
| GetFE(self: ngsolve.comp.FESpace, ei: ngsolve.comp.ElementId) -> object
|
|
| Get the finite element to corresponding element id.
|
| Parameters:
|
| ei : ngsolve.comp.ElementId
| input element id
|
| GetOrder(...)
| GetOrder(self: ngsolve.comp.FESpace, nodeid: ngsolve.comp.NodeId) -> int
|
| return order of node.
| by now, only isotropic order is supported here
|
| GetTrace(...)
| GetTrace(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace, arg1: ngsolve.la.BaseVector, arg2: ngsolve.la.BaseVector, arg3: bool) -> None
|
| GetTraceTrans(...)
| GetTraceTrans(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace, arg1: ngsolve.la.BaseVector, arg2: ngsolve.la.BaseVector, arg3: bool) -> None
|
| HideAllDofs(...)
| HideAllDofs(self: ngsolve.comp.FESpace, component: object = <ngsolve.ngstd.DummyArgument>) -> None
|
| set all visible coupling types to HIDDEN_DOFs (will be overwritten by any Update())
|
| InvM(...)
| InvM(self: ngsolve.comp.FESpace, rho: ngsolve.fem.CoefficientFunction = None) -> BaseMatrix
|
| Mass(...)
| Mass(self: ngsolve.comp.FESpace, rho: ngsolve.fem.CoefficientFunction = None, definedon: Optional[ngsolve.comp.Region] = None) -> BaseMatrix
|
| ParallelDofs(...)
| ParallelDofs(self: ngsolve.comp.FESpace) -> ngsolve.la.ParallelDofs
|
| Return dof-identification for MPI-distributed meshes
|
| Prolongation(...)
| Prolongation(self: ngsolve.comp.FESpace) -> ngmg::Prolongation
|
| Return prolongation operator for use in multi-grid
|
| Range(...)
| Range(self: ngsolve.comp.FESpace, arg0: int) -> ngsolve.la.DofRange
|
| deprecated, will be only available for ProductSpace
|
| SetCouplingType(...)
| SetCouplingType(*args, **kwargs)
| Overloaded function.
|
| 1. SetCouplingType(self: ngsolve.comp.FESpace, dofnr: int, coupling_type: ngsolve.comp.COUPLING_TYPE) -> None
|
|
| Set coupling type of a degree of freedom.
|
| Parameters:
|
| dofnr : int
| input dof number
|
| coupling_type : ngsolve.comp.COUPLING_TYPE
| input coupling type
|
|
|
| 2. SetCouplingType(self: ngsolve.comp.FESpace, dofnrs: ngsolve.ngstd.IntRange, coupling_type: ngsolve.comp.COUPLING_TYPE) -> None
|
|
| Set coupling type for interval of dofs.
|
| Parameters:
|
| dofnrs : Range
| range of dofs
|
| coupling_type : ngsolve.comp.COUPLING_TYPE
| input coupling type
|
| SetDefinedOn(...)
| SetDefinedOn(self: ngsolve.comp.FESpace, region: ngsolve.comp.Region) -> None
|
|
| Set the regions on which the FESpace is defined.
|
| Parameters:
|
| region : ngsolve.comp.Region
| input region
|
| SetOrder(...)
| SetOrder(*args, **kwargs)
| Overloaded function.
|
| 1. SetOrder(self: ngsolve.comp.FESpace, element_type: ngsolve.fem.ET, order: int) -> None
|
|
|
| Parameters:
|
| element_type : ngsolve.fem.ET
| input element type
|
| order : object
| input polynomial order
|
|
| 2. SetOrder(self: ngsolve.comp.FESpace, nodeid: ngsolve.comp.NodeId, order: int) -> None
|
|
|
| Parameters:
|
| nodeid : ngsolve.comp.NodeId
| input node id
|
| order : int
| input polynomial order
|
| SolveM(...)
| SolveM(self: ngsolve.comp.FESpace, vec: ngsolve.la.BaseVector, rho: ngsolve.fem.CoefficientFunction = None, definedon: ngsolve.comp.Region = None) -> None
|
|
| Solve with the mass-matrix. Available only for L2-like spaces.
|
| Parameters:
|
| vec : ngsolve.la.BaseVector
| input right hand side vector
|
| rho : ngsolve.fem.CoefficientFunction
| input CF
|
| TestFunction(...)
| TestFunction(self: ngsolve.comp.FESpace) -> object
|
| Return a proxy to be used as a testfunction for :any:`Symbolic Integrators<symbolic-integrators>`
|
| TnT(...)
| TnT(self: ngsolve.comp.FESpace) -> tuple[object, object]
|
| Return a tuple of trial and testfunction
|
| TraceOperator(...)
| TraceOperator(self: ngsolve.comp.FESpace, tracespace: ngsolve.comp.FESpace, average: bool) -> BaseMatrix
|
| TrialFunction(...)
| TrialFunction(self: ngsolve.comp.FESpace) -> object
|
| Return a proxy to be used as a trialfunction in :any:`Symbolic Integrators<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, space: ngsolve.comp.FESpace) -> bool
|
| __mul__(...)
| __mul__(self: ngsolve.comp.FESpace, arg0: ngsolve.comp.FESpace) -> ngcomp::CompoundFESpace
|
| __pow__(...)
| __pow__(self: ngsolve.comp.FESpace, arg0: int) -> ngcomp::CompoundFESpaceAllSame
|
| __str__(...)
| __str__(self: ngsolve.comp.FESpace) -> str
|
| __timing__(...)
| __timing__(self: ngsolve.comp.FESpace) -> object
|
| ----------------------------------------------------------------------
| Static methods inherited from FESpace:
|
| __special_treated_flags__(...) from builtins.PyCapsule
| __special_treated_flags__() -> dict
|
| ----------------------------------------------------------------------
| Readonly properties inherited from FESpace:
|
| autoupdate
|
| components
| deprecated, will be only available for ProductSpace
|
| couplingtype
|
| dim
| multi-dim of FESpace
|
| globalorder
| query global order of space
|
| is_complex
|
| loembedding
|
| lospace
|
| mesh
| mesh on which the FESpace is created
|
| ndof
| number of degrees of freedom
|
| ndofglobal
| global number of dofs on MPI-distributed mesh
|
| type
| type of finite element space
|
| ----------------------------------------------------------------------
| Data and other attributes inherited from FESpace:
|
| __hash__ = None
|
| ----------------------------------------------------------------------
| Readonly properties inherited from NGS_Object:
|
| __memory__
|
| flags
|
| ----------------------------------------------------------------------
| Data descriptors inherited from NGS_Object:
|
| name
|
| ----------------------------------------------------------------------
| Static 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.
[6]:
u = fes.TrialFunction() # symbolic object
v = fes.TestFunction() # symbolic object
gfu = GridFunction(fes) # solution
Alternately, you can get both the trial and test variables at once:
[7]:
u, v = fes.TnT()
5. Define and assemble linear and bilinear forms:¶
[8]:
a = BilinearForm(fes)
a += grad(u)*grad(v)*dx
a.Assemble()
f = LinearForm(fes)
f += x*v*dx
f.Assemble();
Alternately, we can do one-liners:
[9]:
a = BilinearForm(grad(u)*grad(v)*dx).Assemble()
f = LinearForm(x*v*dx).Assemble()
You can examine the linear system in more detail:
[10]:
print(f.vec)
0.000333333
0.00873506
0.00633333
0.000510926
0.00430724
0.00890537
0.0106804
0.0112774
0.0148637
0.0166495
0.0150628
0.0185037
0.0216167
0.0109997
0.00797031
0.00292185
0.000628106
0.0010144
0.00172207
0.00185741
0.0142294
0.01997
0.0225059
0.020015
0.0311335
0.0259928
0.028898
0.0191106
0.0227498
0.00992941
0.00267856
0.00497134
0.0080662
0.0203373
0.0164825
0.0204557
0.0134827
0.0158788
0.018219
-6.66667e-05
-3.33333e-05
-0.000516153
-0.000566493
-0.0010733
-0.0008
-0.000766667
-5.90186e-05
-1.43029e-05
-7.99565e-05
-0.000293781
-0.000218551
-0.000469969
-0.000398925
-0.000689288
-0.000822998
-0.000483647
-0.000847511
-0.00093399
-0.000895046
-0.000915843
-0.00074941
-0.00109658
-0.00126105
-0.000643908
-0.00140539
-0.00131192
-0.000648404
-0.00121351
-0.00121755
-0.00162374
-0.00149151
-0.00048888
-0.00145143
-0.00110173
-0.000452064
-0.000914624
-0.00088675
-0.000208119
-0.0007941
-0.000543629
-0.000302341
-0.000167379
-2.0832e-05
-6.88223e-05
-8.44745e-05
-2.94073e-05
-0.000116264
-0.000137817
-4.79891e-05
-0.000243969
-0.000195256
-0.000257349
-0.000787648
-0.00044492
-0.000660183
-0.000883158
-0.000835026
-0.000794567
-0.000892677
-0.0010382
-0.000974429
-0.00102675
-0.00114594
-0.000985686
-0.000923147
-0.00108379
-0.0008854
-0.00086299
-0.00104694
-0.000933801
-0.000848623
-0.000866758
-0.000666847
-0.000884373
-0.000729732
-0.000868298
-0.000206371
-0.000281267
-0.000500567
-0.000145299
-0.000258405
-0.000362166
-0.000550055
-0.000437637
-0.00077891
-0.000777094
-0.000731175
-0.000607743
-0.000717949
-0.000729496
-0.000805781
-0.000739587
-0.000678956
[11]:
print(a.mat)
Row 0: 0: 1 4: -0.5 19: -0.5 39: -0.0833333 40: -0.0833333 50: 0.166667
Row 1: 1: 0.828732 7: -0.195774 8: -0.197948 23: -0.43501 41: -0.0360819 42: -0.0364197 43: -0.0656203 59: 0.0687109 61: 0.069411
Row 2: 2: 1 11: -0.5 12: -0.5 44: -0.0833333 45: -0.0833333 69: 0.166667
Row 3: 3: 0.834885 15: -0.164012 16: -0.157951 30: -0.512921 46: -0.0431063 47: -0.0423806 48: -0.0536605 81: 0.0704417 83: 0.0687058
Row 4: 0: -0.5 4: 1.89671 5: -0.546735 19: -0.180532 20: -0.66944 39: 5.83717e-17 40: 0.0833333 49: -0.0544706 50: -0.140436 51: -0.121211 53: 0.145593 91: 0.0871914
Row 5: 4: -0.546735 5: 1.88547 6: -0.352441 20: -0.196284 21: -0.790007 49: -0.0136061 51: 0.104729 52: -0.0481159 53: -0.174674 54: -0.0778481 56: 0.106856 92: 0.10266
Row 6: 5: -0.352441 6: 1.75375 7: -0.265716 21: -0.432048 22: -0.703548 52: -0.0385388 54: 0.097279 55: -0.0537377 56: -0.12226 57: -0.0777552 58: 0.0980238 95: 0.0969894
Row 7: 1: -0.195774 6: -0.265716 7: 1.822 22: -0.405418 23: -0.955095 41: -0.0839333 43: 0.116562 55: -0.0465664 57: 0.0908524 58: -0.119535 59: -0.0536322 98: 0.0962524
Row 8: 1: -0.197948 8: 1.87437 9: -0.341759 23: -1.05108 24: -0.283578 42: -0.0827377 43: 0.115729 60: -0.0333848 61: -0.0468695 62: -0.149403 64: 0.0903445 101: 0.106321
Row 9: 8: -0.341759 9: 1.74315 10: -0.259596 24: -0.651696 25: -0.490101 60: -0.055818 62: 0.112778 63: -0.0349443 64: -0.103699 65: -0.0960639 67: 0.0782102 102: 0.0995372
Row 10: 9: -0.259596 10: 1.76903 11: -0.259654 25: -0.823099 26: -0.426681 63: -0.0694609 65: 0.112727 66: -0.0367441 67: -0.0776353 68: -0.110998 70: 0.0800198 105: 0.102092
Row 11: 2: -0.5 10: -0.259654 11: 1.96958 12: -0.139204 26: -1.07072 44: 0 45: 0.0833333 66: -0.0669124 68: 0.110188 69: -0.194875 70: -0.0664764 72: 0.134742
Row 12: 2: -0.5 11: -0.139204 12: 1.81482 13: -0.275945 26: -0.20179 27: -0.697885 44: 0.0833333 45: 0 69: -0.115666 70: 0.0555335 71: -0.0418546 72: -0.0976603 73: -0.0472897 75: 0.0878455 108: 0.0757584
Row 13: 12: -0.275945 13: 1.79186 14: -0.426433 27: -0.746323 28: -0.343162 71: -0.0571403 73: 0.103131 74: -0.0399256 75: -0.063259 76: -0.138319 78: 0.110998 110: 0.084515
Row 14: 13: -0.426433 14: 1.77214 15: -0.271016 28: -0.405887 29: -0.668803 74: -0.0369995 76: 0.108072 77: -0.0450239 78: -0.137515 79: -0.0758177 80: 0.0901932 112: 0.0970916
Row 15: 3: -0.164012 14: -0.271016 15: 1.81591 29: -0.463066 30: -0.917816 46: -0.0818566 48: 0.109192 77: -0.054447 79: 0.0996163 80: -0.116282 81: -0.0500661 116: 0.0938434
Row 16: 3: -0.157951 16: 1.82092 17: -0.237555 30: -0.936175 31: -0.489235 47: -0.0848367 48: 0.111162 82: -0.0553671 83: -0.0524972 84: -0.110785 86: 0.0949597 119: 0.0973645
Row 17: 16: -0.237555 17: 1.74436 18: -0.336551 31: -0.666886 32: -0.50337 82: -0.0500457 84: 0.0896382 85: -0.0444322 86: -0.0790554 87: -0.117194 90: 0.100524 120: 0.100565
Row 18: 17: -0.336551 18: 1.8616 19: -0.527472 20: -0.241774 32: -0.755808 85: -0.0442896 87: 0.100381 88: -0.015565 89: -0.16959 90: -0.0808224 91: 0.103477 93: 0.106409
Row 19: 0: -0.5 4: -0.180532 18: -0.527472 19: 1.89097 20: -0.682967 39: 0.0833333 40: 6.245e-17 50: -0.143274 51: 0.0900292 88: -0.0538873 89: 0.141799 91: -0.118001
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Row 85: 17: -0.0444322 18: -0.0442896 32: 0.0887218 85: 0.0362034 87: -0.0110724 90: -0.011108
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Row 88: 18: -0.015565 19: -0.0538873 20: 0.0694523 88: 0.0393411 89: -0.0134718 91: -0.00389124
Row 89: 18: -0.16959 19: 0.141799 20: -0.100166 32: 0.127957 88: -0.0134718 89: 0.077513 90: -0.0115697 91: -0.021978 93: -0.0204196
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Row 111: 26: 0.09819 27: -0.104977 28: 0.0842238 35: -0.077437 108: -0.0137312 109: -0.0108162 110: -0.00562801 111: 0.0737236 113: -0.0154279
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Row 113: 27: 0.127669 28: -0.0901223 35: -0.147633 38: 0.110086 110: -0.0164894 111: -0.0154279 113: 0.0755677 115: -0.0204188 130: -0.00710264
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Row 116: 15: 0.0938434 29: -0.0557497 30: -0.127349 31: 0.0892549 80: -0.0177782 81: -0.00568268 116: 0.0742186 117: -0.014059 119: -0.00825474
Row 117: 29: -0.0945602 30: 0.113237 31: -0.102008 36: 0.0833311 116: -0.014059 117: 0.073073 118: -0.0114429 119: -0.0142502 121: -0.00938986
Row 118: 28: 0.0904793 29: -0.108723 31: 0.108477 36: -0.0902331 112: -0.0111154 114: -0.0115045 117: -0.0114429 118: 0.0726635 121: -0.0156763
Row 119: 16: 0.0973645 29: 0.0900197 30: -0.128193 31: -0.059191 83: -0.00654301 84: -0.0177981 116: -0.00825474 117: -0.0142502 119: 0.0739519
Row 120: 17: 0.100565 31: -0.111729 32: -0.0877797 36: 0.0989443 86: -0.0098657 87: -0.0152755 120: 0.0725123 121: -0.0120792 123: -0.0126569
Row 121: 29: 0.100265 31: -0.108168 32: 0.0937796 36: -0.0858763 117: -0.00938986 118: -0.0156763 120: -0.0120792 121: 0.0726108 123: -0.0113657
Row 122: 20: 0.0892148 32: -0.0933717 34: -0.100779 36: 0.104936 93: -0.014545 94: -0.00775875 122: 0.0731559 123: -0.0106497 127: -0.0155842
Row 123: 31: 0.0960903 32: -0.0863352 34: 0.0834713 36: -0.0932263 120: -0.0126569 121: -0.0113657 122: -0.0106497 123: 0.0725538 127: -0.0102181
Row 124: 21: 0.0958583 33: -0.113287 34: -0.0852611 38: 0.10269 96: -0.00713071 97: -0.0168339 124: 0.0732984 126: -0.0141846 128: -0.0114879
Row 125: 24: 0.0711795 33: -0.0446607 37: -0.106982 38: 0.0804635 103: -0.0114075 104: -0.00638735 125: 0.0761401 126: -0.0153381 132: -0.00477781
Row 126: 33: -0.11397 34: 0.0989745 37: 0.133086 38: -0.118091 124: -0.0141846 125: -0.0153381 126: 0.0742808 128: -0.0105591 132: -0.0179334
Row 127: 32: 0.103209 34: -0.103379 36: -0.113503 38: 0.113673 122: -0.0155842 123: -0.0102181 127: 0.0731092 128: -0.0127917 131: -0.0156265
Row 128: 33: 0.0881878 34: -0.0751922 36: 0.0841228 38: -0.0971183 124: -0.0114879 126: -0.0105591 127: -0.0127917 128: 0.0728888 131: -0.00823901
Row 129: 25: 0.0935649 35: -0.0644794 37: -0.120086 38: 0.0910005 106: -0.0173819 107: -0.00600932 129: 0.0736918 130: -0.0126396 132: -0.0101105
Row 130: 28: 0.0704143 35: -0.0958469 37: 0.104402 38: -0.0789689 113: -0.00710264 115: -0.0105009 129: -0.0126396 130: 0.0742333 132: -0.0134608
Row 131: 28: 0.0850283 34: 0.0954622 36: -0.083436 38: -0.0970544 114: -0.00863707 115: -0.01262 127: -0.0156265 128: -0.00823901 131: 0.0732047
Row 132: 33: 0.0908447 35: 0.0942853 37: -0.125577 38: -0.0595534 125: -0.00477781 126: -0.0179334 129: -0.0101105 130: -0.0134608 132: 0.0742602
6. Solve the system:¶
[12]:
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:
[13]:
print(gfu.vec)
0
0
0
0.0923179
0
0
0
0
0
0
0
0
0.0578988
0.0863391
0.0954151
0.0945036
0.0888226
0.0780489
0.0595559
0.0331334
0.043177
0.0384731
0.0312766
0.0180508
0.0379191
0.0426386
0.0472136
0.0758445
0.0892927
0.0932644
0.0919339
0.0863
0.0722526
0.0576401
0.0666805
0.0710587
0.0846294
0.0618883
0.0766818
0
-0.00563337
0
0
0.0203536
0
-0.0350728
0.00280938
-0.00728756
-0.00137235
0
-0.00966674
-0.0121644
0
-0.0238901
-0.0131488
0
-0.0161892
-0.0200118
-0.00657505
-0.0212905
0
-0.0255408
-0.00925383
0
-0.0375247
-0.0131487
0
-0.0261275
-0.0183435
-0.0333457
-0.0247246
-0.0242762
-0.00473494
-0.0120835
-0.0145349
-0.00640155
-0.0108854
-0.00549118
-0.00937973
-0.00705122
-0.0052206
-0.00304656
-0.00758425
0.00107111
0.00110145
-0.00824163
0.000388213
0.00186718
-0.0082305
-0.000562953
0.00136394
0.00464277
-0.00573891
-0.0141064
-0.0092274
-0.00723695
-0.0140782
-0.012669
-0.0071093
-0.0248443
-0.00762662
-0.00336514
-0.00878309
-0.018877
-0.0045665
-0.00280081
-0.0126613
-0.0165067
-0.0151556
-0.0190153
-0.0164166
-0.00385688
-0.0106348
-0.0142227
-0.0122429
-0.00847428
-0.00104741
-0.00422534
-0.00710203
-0.00456872
-0.00792875
-0.00279066
-0.00418665
-0.00609652
-0.00917816
-0.0111537
-0.00714295
-0.00950903
-0.0114547
-0.00372816
-0.0141414
-0.00887479
-0.00923917
You can see the zeros coming from the zero boundary conditions.
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 module in a text editor and call python on the command line. (A script of the above is provided in
poisson.py.)python3 poisson.pyIf you want the Netgen GUI, then use
netgenon the command line:netgen poisson.pyYou can then ask for a python shell from the GUI’s menu options (Solve -> Python shell).