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]:
import netgen.gui
from ngsolve import *
from netgen.geom2d import unit_square
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
maxhflag.The mesh can be viewed by switching to the
Meshtab in the Netgen GUI.
3. Declare a finite element space:¶
[3]:
fes = H1(mesh, order=2, dirichlet="bottom|right")
fes.ndof # number of unknowns in this space
[3]:
133
Python’s help system displays further documentation.
[4]:
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.
| low_order_space: bool = True
| Generate a lowest order space together with the high-order space,
| needed for some preconditioners.
| 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,
| VARIBLE ... use an individual order for each edge, face and cell,
| OLDSTYLE .. as it used to be for the last decade
| 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
|
| 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, autoupdate: bool = False, **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
|
| Elements(...)
| 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,...
|
| FinalizeUpdate(...)
| FinalizeUpdate(self: ngsolve.comp.FESpace) -> None
|
| finalize update
|
| FreeDofs(...)
| FreeDofs(self: ngsolve.comp.FESpace, coupling: bool = False) -> 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.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:
|
| 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__
|
| ----------------------------------------------------------------------
| 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.
[5]:
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:
[6]:
u, v = fes.TnT()
5. Define and assemble linear and bilinear forms:¶
[7]:
a = BilinearForm(fes, symmetric=True)
a += grad(u)*grad(v)*dx
a.Assemble()
f = LinearForm(fes)
f += x*v*dx
f.Assemble()
[7]:
<ngsolve.comp.LinearForm at 0x7f2133b63d30>
You can examine the linear system in more detail:
[8]:
print(f.vec)
0.000333333
0.00888574
0.00633333
0.00074628
0.00399953
0.00769352
0.0104008
0.0115294
0.0150217
0.0167909
0.015258
0.0182768
0.0211091
0.0106045
0.00711336
0.00320035
0.000735854
0.000876249
0.000529403
0.00282218
0.0124495
0.0182096
0.0224435
0.0204234
0.0317338
0.0270543
0.0291603
0.0187131
0.0217393
0.00854715
0.00505989
0.0038006
0.00883394
0.0205505
0.0163797
0.0213793
0.0150785
0.0170279
0.0191555
-6.66667e-05
-3.33333e-05
-0.000526369
-0.000575033
-0.00109143
-0.0008
-0.000766667
-8.08556e-05
-2.24397e-05
-0.000120589
-0.000249038
-0.000218716
-0.000440093
-0.000380259
-0.000572665
-0.000711098
-0.000487162
-0.00080946
-0.000916707
-0.000914728
-0.000944161
-0.000759221
-0.00110619
-0.00127157
-0.000646306
-0.0014197
-0.00131979
-0.000648788
-0.00123697
-0.0012392
-0.0015957
-0.00146355
-0.000469258
-0.0014082
-0.00106589
-0.000437144
-0.000880349
-0.000856527
-0.000190932
-0.000699086
-0.000461922
-0.000315546
-0.000220954
-1.64401e-05
-9.82846e-05
-8.35917e-05
-2.64702e-05
-9.73025e-05
-0.000122662
-2.64702e-05
-0.000105881
-0.000353264
-0.00021487
-0.00066398
-0.000468351
-0.00059592
-0.000843865
-0.000796408
-0.000722285
-0.000922806
-0.00105484
-0.000955488
-0.00104202
-0.00117171
-0.00100951
-0.000963748
-0.00114996
-0.000965717
-0.00092025
-0.00104152
-0.00098591
-0.000825307
-0.000865806
-0.000574259
-0.000880372
-0.000710639
-0.000885225
-0.000308496
-0.000477574
-0.000191583
-0.00032424
-0.000250767
-0.000328592
-0.000559968
-0.00047995
-0.000773918
-0.00082872
-0.000774775
-0.000675938
-0.000761632
-0.000801479
-0.000841885
-0.000775647
-0.000743459
[9]:
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: 0.828441 7: -0.208968 8: -0.209694 23: -0.40978 41: -0.0340869 42: -0.0342097 43: -0.069777 59: 0.0689149 61: 0.0691587
Row 2: 2: 1 11: -0.5 12: -0.5 44: -0.0833333 45: -0.0833333 69: 0.166667
Row 3: 3: 0.870927 15: -0.344377 16: -0.336878 30: -0.189673 46: -0.0173948 47: -0.0142173 48: -0.113542 81: 0.074791 83: 0.0703636
Row 4: 0: -0.5 4: 1.88347 5: -0.397227 19: -0.177276 20: -0.808965 39: 0 40: 0.0833333 49: -0.0535736 50: -0.164587 51: -0.0957505 53: 0.119778 90: 0.1108
Row 5: 4: -0.397227 5: 1.77564 6: -0.326749 20: -0.345183 21: -0.706484 49: -0.028366 51: 0.0945705 52: -0.0526905 53: -0.131261 54: -0.0836226 56: 0.107149 92: 0.0942211
Row 6: 5: -0.326749 6: 1.7517 7: -0.273913 21: -0.432506 22: -0.718531 52: -0.0380314 54: 0.0924896 55: -0.0573097 56: -0.116904 57: -0.0797051 58: 0.102962 95: 0.0964985
Row 7: 1: -0.208968 6: -0.273913 7: 1.82559 22: -0.374026 23: -0.968685 41: -0.0835417 43: 0.11837 55: -0.0420363 57: 0.0876885 58: -0.123558 59: -0.0551294 98: 0.0982072
Row 8: 1: -0.209694 8: 1.8809 9: -0.350939 23: -1.05698 24: -0.263291 42: -0.0831255 43: 0.118074 60: -0.0322109 61: -0.0466199 62: -0.151527 64: 0.0907008 101: 0.104708
Row 9: 8: -0.350939 9: 1.74429 10: -0.262838 24: -0.64648 25: -0.484029 60: -0.0557926 62: 0.114282 63: -0.0342301 64: -0.104931 65: -0.0957605 67: 0.0780365 102: 0.0983955
Row 10: 9: -0.262838 10: 1.77715 11: -0.267845 25: -0.84644 26: -0.400025 63: -0.0697744 65: 0.113581 66: -0.0327463 67: -0.0777309 68: -0.11594 70: 0.0773871 105: 0.105223
Row 11: 2: -0.5 10: -0.267845 11: 1.9829 12: -0.115438 26: -1.09961 44: 0 45: 0.0833333 66: -0.0708467 68: 0.115488 69: -0.195755 70: -0.0638806 72: 0.131662
Row 12: 2: -0.5 11: -0.115438 12: 1.81266 13: -0.253469 26: -0.239874 27: -0.703881 44: 0.0833333 45: 0 69: -0.11965 70: 0.0555561 71: -0.0432149 72: -0.0933383 73: -0.0459076 75: 0.0854598 108: 0.0777613
Row 13: 12: -0.253469 13: 1.79007 14: -0.404897 27: -0.775929 28: -0.355771 71: -0.0598976 73: 0.102142 74: -0.0406718 75: -0.0608683 76: -0.136907 78: 0.108155 110: 0.0880473
Row 14: 13: -0.404897 14: 1.83152 15: -0.218833 28: -0.309609 29: -0.898185 74: -0.0388314 76: 0.106314 77: -0.0683139 78: -0.148866 79: -0.0492423 80: 0.104786 112: 0.0941536
Row 15: 3: -0.344377 14: -0.218833 15: 1.78023 29: -0.432178 30: -0.784846 46: -0.0795022 48: 0.136898 77: -0.0424951 79: 0.0789672 80: -0.0877775 81: -0.0869308 116: 0.0808401
Row 16: 3: -0.336878 16: 1.81347 17: -0.20561 30: -0.908731 31: -0.36225 47: -0.0873491 48: 0.143495 82: -0.033994 83: -0.0825273 84: -0.0983745 86: 0.0682624 118: 0.0904872
Row 17: 16: -0.20561 17: 1.94288 18: -0.368349 31: -1.16439 32: -0.204537 82: -0.0846663 84: 0.118935 85: -0.0154075 86: -0.0529503 87: -0.17079 89: 0.076799 120: 0.12808
Row 18: 17: -0.368349 18: 2.04376 19: -0.553239 32: -1.12218 85: -0.0781072 87: 0.139499 88: -0.108922 89: -0.153598 91: 0.201129
Row 19: 0: -0.5 4: -0.177276 18: -0.553239 19: 1.87497 20: -0.545937 32: -0.098519 39: 0.0833333 40: 0 50: -0.124342 51: 0.0705543 88: 0.0154075 89: 0.076799 90: -0.0613734 91: -0.142188 93: 0.0818086
Row 20: 4: -0.808965 5: -0.345183 19: -0.545937 20: 3.54246 21: -0.647227 32: -0.584699 34: -0.610448 49: 0.0819397 50: 0.122262 51: -0.0693744 53: -0.10714 54: 0.082731 90: -0.146695 91: 0.115423 92: -0.077413 93: -0.103474 94: -0.0863131 97: 0.102553 122: 0.0855011
Row 21: 5: -0.706484 6: -0.432506 20: -0.647227 21: 3.50609 22: -0.571505 33: -0.714154 34: -0.43421 52: 0.0907219 53: 0.118623 54: -0.091598 56: -0.102619 57: 0.0839811 92: -0.107223 94: 0.0964712 95: -0.0880292 96: -0.0755245 97: -0.119354 100: 0.0992989 124: 0.0952513
Row 22: 6: -0.718531 7: -0.374026 21: -0.571505 22: 3.51428 23: -0.708085 24: -0.536887 33: -0.60525 55: 0.099346 56: 0.112374 57: -0.0919644 58: -0.111917 59: 0.0749089 95: -0.107744 96: 0.0906212 98: -0.0555055 99: -0.118983 100: -0.0995997 101: 0.0986109 103: 0.109853
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Row 116: 15: 0.0808401 29: -0.0832396 30: -0.0628419 36: 0.0652415 80: -0.0128264 81: -0.00738366 116: 0.0775466 117: -0.00288412 119: -0.0134262
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Row 119: 29: 0.150651 30: -0.180835 31: 0.157852 36: -0.127668 116: -0.0134262 117: -0.0242364 118: -0.0184907 119: 0.0811814 121: -0.0209724
Row 120: 17: 0.12808 31: -0.195221 32: -0.0398024 36: 0.106943 86: -0.00467048 87: -0.0273496 120: 0.0803181 121: -0.00528012 123: -0.0214556
Row 121: 30: 0.0885758 31: -0.131876 32: 0.0691071 36: -0.0258068 118: -0.00117156 119: -0.0209724 120: -0.00528012 121: 0.0793671 123: -0.0119967
Row 122: 20: 0.0855011 32: -0.0718724 34: -0.0897809 36: 0.0761522 93: -0.0133732 94: -0.00800211 122: 0.0737609 123: -0.00907206 127: -0.00996599
Row 123: 31: 0.133809 32: -0.120182 34: 0.108484 36: -0.122111 120: -0.0214556 121: -0.0119967 122: -0.00907206 123: 0.0758193 127: -0.0180489
Row 124: 21: 0.0952513 33: -0.105243 34: -0.083047 38: 0.0930384 96: -0.00755045 97: -0.0162624 124: 0.0730955 126: -0.0132113 128: -0.0100483
Row 125: 24: 0.072687 33: -0.0419266 37: -0.108677 38: 0.0779166 103: -0.0123289 104: -0.00584281 125: 0.076319 126: -0.0148403 132: -0.00463885
Row 126: 33: -0.126801 34: 0.104656 37: 0.134351 38: -0.112206 124: -0.0132113 125: -0.0148403 126: 0.0744391 128: -0.0129528 132: -0.0187475
Row 127: 32: 0.11206 34: -0.140852 36: -0.101772 38: 0.130564 122: -0.00996599 123: -0.0180489 127: 0.0748865 128: -0.015477 131: -0.0171641
Row 128: 33: 0.0920044 34: -0.0724453 36: 0.0825421 38: -0.102101 124: -0.0100483 126: -0.0129528 127: -0.015477 128: 0.074012 131: -0.00515852
Row 129: 25: 0.0933844 35: -0.0640386 37: -0.118715 38: 0.0893687 106: -0.0175598 107: -0.00578633 129: 0.0738082 130: -0.0121189 132: -0.0102233
Row 130: 28: 0.0767268 35: -0.103918 37: 0.104 38: -0.0768082 113: -0.00708319 115: -0.0120985 129: -0.0121189 130: 0.0735645 132: -0.0138811
Row 131: 28: 0.0848918 34: 0.0892903 36: -0.0657272 38: -0.108455 114: -0.00994967 115: -0.0112733 127: -0.0171641 128: -0.00515852 131: 0.0741883
Row 132: 33: 0.0935455 35: 0.0964175 37: -0.130514 38: -0.0594486 125: -0.00463885 126: -0.0187475 129: -0.0102233 130: -0.0138811 132: 0.0744499
6. Solve the system:¶
[10]:
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:
[11]:
print(gfu.vec)
0
0
0
0.0923044
0
0
0
0
0
0
0
0
0.0578972
0.0863393
0.0954128
0.0944891
0.0888235
0.0780439
0.059628
0.0330628
0.0374521
0.0370446
0.0317995
0.0185123
0.0383632
0.0427969
0.047351
0.0760518
0.0895602
0.0939297
0.0914143
0.0833171
0.0654761
0.0574569
0.0647326
0.0723792
0.0852186
0.0626708
0.0776805
0
-0.00576774
0
0
0.0208108
0
-0.0350847
0.00263379
-0.00729372
-0.0027615
0
-0.00993549
-0.00711876
0
-0.0174982
-0.0120617
0
-0.0150936
-0.0195578
-0.00638541
-0.0213554
0
-0.0254715
-0.00955261
0
-0.0381111
-0.0132127
0
-0.0262138
-0.0182002
-0.0333693
-0.0245117
-0.0242838
-0.00451911
-0.0119297
-0.014536
-0.00612786
-0.0105674
-0.00558557
-0.00891829
-0.00493114
-0.0051195
-0.00519767
-0.007602
0.00247896
-0.00133557
-0.00800354
0.00157024
-0.00177994
-0.00829596
0.00374357
0.00436653
-0.00759115
-0.00373981
-0.0120787
-0.0104187
-0.00708368
-0.0136752
-0.0120743
-0.00778966
-0.0256197
-0.0072408
-0.00331602
-0.00900117
-0.0206949
-0.0050238
-0.00300073
-0.0140407
-0.0179806
-0.0153171
-0.020859
-0.0164647
-0.00404961
-0.00810738
-0.0130797
-0.0135373
-0.00892006
-0.00206078
-0.00747051
-0.00532125
-0.00675915
-0.00966483
-0.00121964
-0.00276467
-0.0111685
-0.00868831
-0.0118313
-0.00814322
-0.0124053
-0.0120342
-0.00417412
-0.0154546
-0.00866456
-0.010265
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).