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]:
(38, 54)
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]:
129
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__(...) method of builtins.PyCapsule instance
| __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
| exclude 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__(...) method of builtins.PyCapsule instance
| __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) class method of pybind11_builtins.pybind11_object
| 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
0.00633333
0.00633333
0.000529288
0.00423206
0.00879483
0.0115687
0.0176999
0.0273014
0.0166436
0.0151442
0.0182045
0.0210073
0.0107111
0.00781528
0.00306235
0.00067471
0.00103167
0.00166604
0.00181674
0.0137015
0.0205046
0.0304556
0.0291057
0.0260001
0.0287722
0.0186625
0.0219344
0.0100352
0.00291583
0.00518005
0.00777967
0.0205471
0.0158929
0.0207949
0.0129617
0.0158859
0.0179665
-6.66667e-05
-3.33333e-05
-0.000766667
-0.0008
-0.0008
-0.000766667
-6.06392e-05
-1.49949e-05
-8.31522e-05
-0.000289108
-0.000214565
-0.000459295
-0.000398603
-0.000676002
-0.000812886
-0.000522023
-0.000916349
-0.00105268
-0.00172646
-0.00143407
-0.000777223
-0.00178112
-0.00160576
-0.000646778
-0.00139999
-0.00128445
-0.000648431
-0.00122331
-0.00122479
-0.00158722
-0.00145468
-0.000467351
-0.00139751
-0.00106135
-0.000445345
-0.000885342
-0.000870842
-0.000215685
-0.00076362
-0.000534472
-0.000320341
-0.000177293
-2.19785e-05
-7.42829e-05
-9.11566e-05
-2.9417e-05
-0.000119731
-0.000138374
-4.68667e-05
-0.000234449
-0.00018908
-0.000250255
-0.000763332
-0.000417469
-0.00062207
-0.000994616
-0.000858919
-0.000770058
-0.00135916
-0.0010122
-0.00108461
-0.00096346
-0.000872767
-0.0011199
-0.000934733
-0.000855051
-0.00103163
-0.000963124
-0.000829911
-0.00085758
-0.00063321
-0.000865444
-0.000687957
-0.000829446
-0.00023004
-0.000301583
-0.000494185
-0.0001632
-0.000257852
-0.000363996
-0.000519229
-0.000426959
-0.000777698
-0.000773376
-0.000750942
-0.00057903
-0.000701523
-0.000764691
-0.000809458
-0.000691204
-0.000705732
[11]:
print(a.mat)
Row 0: 0: 1 4: -0.5 19: -0.5 38: -0.0833333 39: -0.0833333 48: 0.166667
Row 1: 1: 1 7: -0.5 8: -0.5 40: -0.0833333 41: -0.0833333 56: 0.166667
Row 2: 2: 1 11: -0.5 12: -0.5 42: -0.0833333 43: -0.0833333 67: 0.166667
Row 3: 3: 0.831314 15: -0.17846 16: -0.173448 29: -0.479406 44: -0.040286 45: -0.039615 46: -0.0586512 79: 0.0700292 81: 0.068523
Row 4: 0: -0.5 4: 1.89829 5: -0.540848 19: -0.168769 20: -0.688673 38: 5.83717e-17 39: 0.0833333 47: -0.0562171 48: -0.141895 49: -0.118269 51: 0.146359 89: 0.0866899
Row 5: 4: -0.540848 5: 1.88645 6: -0.354514 20: -0.188208 21: -0.802875 47: -0.0128244 49: 0.102966 50: -0.0499561 51: -0.173998 52: -0.0776293 54: 0.109042 90: 0.1024
Row 6: 5: -0.354514 6: 1.74447 7: -0.307659 21: -0.489892 22: -0.592403 50: -0.0366167 52: 0.0957024 53: -0.0414522 54: -0.116367 55: -0.0963085 57: 0.0927287 93: 0.102314
Row 7: 1: -0.5 6: -0.307659 7: 1.90404 8: -0.20374 22: -0.892641 40: 0 41: 0.0833333 53: -0.051968 55: 0.103244 56: -0.180139 57: -0.0852332 59: 0.130762
Row 8: 1: -0.5 7: -0.20374 8: 1.79311 9: -0.389613 22: -0.247724 23: -0.452028 40: 0.0833333 41: 0 56: -0.111302 57: 0.0619254 58: -0.0211183 59: -0.0881764 60: -0.078254 62: 0.0860538 96: 0.0675384
Row 9: 8: -0.389613 9: 1.76257 10: -0.271361 23: -0.692568 24: -0.409029 58: -0.0647632 60: 0.129699 61: -0.03061 62: -0.102497 63: -0.0958916 65: 0.0758369 98: 0.0882263
Row 10: 9: -0.271361 10: 1.78763 11: -0.270137 24: -0.87395 25: -0.372178 61: -0.0733159 63: 0.118543 64: -0.031622 65: -0.0756344 66: -0.117365 68: 0.0766449 101: 0.10275
Row 11: 2: -0.5 10: -0.270137 11: 1.98752 12: -0.108449 25: -1.10893 42: 0 43: 0.0833333 64: -0.07203 66: 0.117053 67: -0.196125 68: -0.0630977 70: 0.130867
Row 12: 2: -0.5 11: -0.108449 12: 1.81026 13: -0.251506 25: -0.253079 26: -0.697226 42: 0.0833333 43: 0 67: -0.12082 68: 0.0555613 69: -0.0431575 70: -0.0911215 71: -0.046611 73: 0.0850753 104: 0.07774
Row 13: 12: -0.251506 13: 1.80179 14: -0.417484 26: -0.798894 27: -0.333901 69: -0.0603628 71: 0.10228 72: -0.0389519 73: -0.058616 74: -0.142367 76: 0.108533 106: 0.0894845
Row 14: 13: -0.417484 14: 1.76811 15: -0.286119 27: -0.397757 28: -0.666748 72: -0.0390126 74: 0.108593 75: -0.0488037 76: -0.131902 77: -0.0749666 78: 0.0964902 108: 0.0896011
Row 15: 3: -0.17846 14: -0.286119 15: 1.82432 28: -0.407682 29: -0.952064 44: -0.0820544 46: 0.111798 75: -0.0478511 77: 0.0955376 78: -0.124309 79: -0.0498391 112: 0.0967187
Row 16: 3: -0.173448 16: 1.8265 17: -0.252776 29: -0.969793 30: -0.430482 45: -0.0846322 46: 0.11354 80: -0.0478231 81: -0.0528319 82: -0.119129 84: 0.0899525 115: 0.100924
Row 17: 16: -0.252776 17: 1.75225 18: -0.338143 30: -0.71401 31: -0.447317 80: -0.0548032 82: 0.0969326 83: -0.039654 84: -0.0770282 85: -0.120556 88: 0.0960112 116: 0.0990973
Row 18: 17: -0.338143 18: 1.87019 19: -0.523655 20: -0.213918 31: -0.794476 83: -0.0490532 85: 0.10541 86: -0.014548 87: -0.170635 88: -0.0774621 89: 0.101824 91: 0.104464
Row 19: 0: -0.5 4: -0.168769 18: -0.523655 19: 1.89344 20: -0.701018 38: 0.0833333 39: 6.245e-17 48: -0.144439 49: 0.0892334 86: -0.055731 87: 0.143007 89: -0.115404
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Row 117: 28: 0.0930433 30: -0.0992521 31: 0.0893782 35: -0.0831694 113: -0.00982642 114: -0.0134344 116: -0.0109659 117: 0.0724293 119: -0.0113786
Row 118: 20: 0.0869532 31: -0.0875396 33: -0.0977775 35: 0.0983639 91: -0.0140519 92: -0.00768643 118: 0.0729761 119: -0.0103925 123: -0.0141985
Row 119: 30: 0.100875 31: -0.0921121 33: 0.0881677 35: -0.0969304 116: -0.0138401 117: -0.0113786 118: -0.0103925 119: 0.072425 123: -0.0116494
Row 120: 21: 0.0962176 32: -0.109837 33: -0.0853952 37: 0.099015 94: -0.00753034 95: -0.0165241 120: 0.0731081 122: -0.0138185 124: -0.0109353
Row 121: 23: 0.0780061 32: -0.0414894 36: -0.117275 37: 0.080758 99: -0.0138982 100: -0.00560337 121: 0.075809 122: -0.0154205 128: -0.00476897
Row 122: 32: -0.117138 33: 0.100991 36: 0.133103 37: -0.116956 120: -0.0138185 121: -0.0154205 122: 0.0742278 124: -0.0114293 128: -0.0178552
Row 123: 31: 0.103391 33: -0.101202 35: -0.118433 37: 0.116244 118: -0.0141985 119: -0.0116494 123: 0.0729978 124: -0.0154099 127: -0.0136511
Row 124: 32: 0.0894585 33: -0.0765031 35: 0.0924253 37: -0.105381 120: -0.0109353 122: -0.0114293 123: -0.0154099 124: 0.0729405 127: -0.00769644
Row 125: 24: 0.091449 34: -0.0610592 36: -0.120117 37: 0.0897272 102: -0.0172502 103: -0.00561208 125: 0.0738938 126: -0.0127791 128: -0.00965271
Row 126: 27: 0.0761289 34: -0.102139 36: 0.106515 37: -0.0805053 109: -0.00734725 111: -0.011685 125: -0.0127791 126: 0.0736078 128: -0.0138498
Row 127: 27: 0.0771363 33: 0.0853902 35: -0.0741181 37: -0.0884084 110: -0.008451 111: -0.0108331 123: -0.0136511 124: -0.00769644 127: 0.0738011
Row 128: 32: 0.0904968 34: 0.0940099 36: -0.12682 37: -0.0576867 121: -0.00476897 122: -0.0178552 125: -0.00965271 126: -0.0138498 128: 0.0743263
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
0.0923181
0
0
0
0
0
0
0
0
0.0578964
0.0863391
0.0954146
0.0945008
0.0888215
0.0780489
0.0595585
0.0331336
0.0426142
0.0384925
0.0322658
0.0404939
0.0430216
0.0473617
0.076118
0.0895333
0.0932199
0.0919131
0.0857457
0.0712628
0.0581085
0.0661322
0.0723769
0.0842824
0.0630681
0.0775393
0
-0.00562995
0
0
0
-0.035088
0.002793
-0.00729426
-0.00144059
0
-0.00965991
-0.0118565
0
-0.023484
-0.0130763
0
-0.016122
-0.0242459
-0.0872423
-0.00749163
0
-0.053399
-0.0070418
0
-0.0361924
-0.0128913
0
-0.0258153
-0.0181385
-0.033376
-0.0244207
-0.0242838
-0.00446541
-0.0119724
-0.01453
-0.00607083
-0.0110517
-0.00550777
-0.0090215
-0.00690385
-0.00564518
-0.00320331
-0.00759005
0.00119038
0.000777561
-0.0082326
0.000951081
0.00144642
-0.00822658
-0.00086348
0.0018314
0.00459756
-0.00544951
-0.0133503
-0.0087701
-0.010031
-0.013985
-0.0122591
-0.0222892
-0.0067813
-0.00811706
-0.0208728
-0.00480275
-0.00284346
-0.0139733
-0.0176085
-0.015435
-0.0205775
-0.0168732
-0.00387324
-0.00933138
-0.0134445
-0.0124145
-0.00837241
-0.00130944
-0.00480625
-0.00707163
-0.00509836
-0.00790602
-0.00259686
-0.00365637
-0.00637113
-0.0091458
-0.0114363
-0.00754309
-0.00951891
-0.011657
-0.00386109
-0.0148107
-0.00755826
-0.00980606
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).