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3     % Copyright © 2006 by ACcESS MNRF
4     % \url{http://www.access.edu.au
5     % Primary Business: Queensland, Australia.
6     % Licensed under the Open Software License version 3.0
7     % http://www.opensource.org/licenses/osl-3.0.php
8     %
9 jgs 102
10 gross 625
11 gross 599 \chapter{The module \linearPDEs}
12 jgs 102
13     \declaremodule{extension}{linearPDEs} \modulesynopsis{Linear partial pifferential equation handler}
14 gross 599 The module \linearPDEs provides an interface to define and solve linear partial
15     differential equations within \escript. \linearPDEs does not provide any
16 jgs 107 solver capabilities in itself but hands the PDE over to
17 jgs 102 the PDE solver library defined through the \Domain of the PDE.
18     The general interface is provided through the \LinearPDE class. The
19     \AdvectivePDE which is derived from the \LinearPDE class
20     provides an interface to PDE dominated by its advective terms. The \Poisson
21     class which is also derived form the \LinearPDE class should be used
22     to define the Poisson equation \index{Poisson}.
23    
24     \section{\LinearPDE Class}
25     \label{SEC LinearPDE}
26    
27     The \LinearPDE class is used to define a general linear, steady, second order PDE
28     for an unknown function $u$ on a given $\Omega$ defined through a \Domain object.
29     In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes
30     the outer normal field on $\Gamma$.
31    
32 jgs 107 For a single PDE with a solution with a single component the linear PDE is defined in the
33 jgs 102 following form:
34     \begin{equation}\label{LINEARPDE.SINGLE.1}
35 gross 625 -(A\hackscore{jl} u\hackscore{,l})\hackscore{,j}+(B\hackscore{j} u)\hackscore{,j}+C\hackscore{l} u\hackscore{,l}+D u =-X\hackscore{j,j}+Y \; .
36 jgs 102 \end{equation}
37     $u_{,j}$ denotes the derivative of $u$ with respect to the $j$-th spatial direction. Einstein's summation convention, ie. summation over indexes appearing twice in a term of a sum is performed, is used.
38     The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the
39     \Function on the PDE or objects that can be converted into such \Data objects.
40     $A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar.
41     The following natural
42     boundary conditions are considered \index{boundary condition!natural} on $\Gamma$:
43     \begin{equation}\label{LINEARPDE.SINGLE.2}
44     n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;.
45     \end{equation}
46     Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are
47     each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the
48     solution at certain locations in the domain. They have the form
49     \begin{equation}\label{LINEARPDE.SINGLE.3}
50     u=r \mbox{ where } q>0
51     \end{equation}
52     $r$ and $q$ are each \Scalar where $q$ is the characteristic function
53     \index{characteristic function} defining where the constraint is applied.
54     The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1}
55 gross 625 or \eqn{LINEARPDE.SINGLE.2}.
56    
57 jgs 102 For a system of PDEs and a solution with several components the PDE has the form
58     \begin{equation}\label{LINEARPDE.SYSTEM.1}
59 gross 625 -(A\hackscore{ijkl} u\hackscore{k,l}){,j}+(B\hackscore{ijk} u\hackscore{k})\hackscore{,j}+C\hackscore{ikl} u\hackscore{k,l}+D\hackscore{ik} u\hackscore{k} =-X\hackscore{ij,j}+Y\hackscore{i} \; .
60 jgs 102 \end{equation}
61     $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne.
62     The natural boundary conditions \index{boundary condition!natural} take the form:
63     \begin{equation}\label{LINEARPDE.SYSTEM.2}
64 gross 625 n\hackscore{j}(A\hackscore{ijkl} u\hackscore{k,l}+B\hackscore{ijk} u\hackscore{k})+d\hackscore{ik} u\hackscore{k}=n\hackscore{j}X\hackscore{ij}+y\hackscore{i} \;.
65 jgs 102 \end{equation}
66     The coefficient $d$ is a \RankTwo and $y$ is a
67     \RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form
68     \begin{equation}\label{LINEARPDE.SYSTEM.3}
69     u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0
70     \end{equation}
71 gross 625 $r$ and $q$ are each \RankOne. Notice that not necessarily all components must
72     have a constraint at all locations.
73    
74 jgs 102 \LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$
75     in the domain $\Omega$. To specify the conditions across the discontinuity we are using the
76     generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution
77     defined as
78     \begin{equation}\label{LINEARPDE.SYSTEM.5}
79     J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij}
80     \end{equation}
81     For the case of single solution component and single PDE $J$ is defined
82     \begin{equation}\label{LINEARPDE.SINGLE.5}
83     J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j}
84     \end{equation}
85     In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the
86     discontinuity pointing from side 0 towards side 1. For a system of PDEs
87     the contact condition takes the form
88     \begin{equation}\label{LINEARPDE.SYSTEM.6}
89     n\hackscore{j} J^{0}\hackscore{ij}=n\hackscore{j} J^{1}\hackscore{ij}=y^{contact}\hackscore{i} - d^{contact}\hackscore{ik} [u]\hackscore{k} \; .
90     \end{equation}
91     where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the
92     discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference
93     of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$.
94     The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a
95     \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne.
96     In case of a single PDE and a single component solution the contact condition takes the form
97     \begin{equation}\label{LINEARPDE.SINGLE.6}
98     n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u]
99     \end{equation}
100     In this case the the coefficient $d^{contact}$ and $y^{contact}$ are eaach \Scalar
101     both in the \FunctionOnContactZero or \FunctionOnContactOne.
102 gross 625
103     The PDE is symmetrical \index{symmetrical} if
104     \begin{equation}\label{LINEARPDE.SINGLE.4}
105     A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j}
106     \end{equation}
107     The system of PDEs is symmetrical \index{symmetrical} if
108     \begin{eqnarray}
109     \label{LINEARPDE.SYSTEM.4}
110     A\hackscore{ijkl}=A\hackscore{klij} \\
111     B\hackscore{ijk}=C\hackscore{kij} \\
112     D\hackscore{ik}=D\hackscore{ki} \\
113     d\hackscore{ik}=d\hackscore{ki} \\
114     d^{contact}\hackscore{ik}=d^{contact}\hackscore{ki}
115     \end{eqnarray}
116     Note that different from the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$
117     have to be inspected.
118    
119     \section{\LinearPDE class}
120     This is the general class to define a linear PDE in \escript. We list a selction of the most
121     important methods of the class only and refer to reference guide \ReferenceGuide for a complete list.
122    
123 jgs 102 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0}
124     opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations}
125     and \var{numSolutions} gives the number of equations and the number of solutiopn components.
126     If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations
127     and the number solutions, respctively, stay undefined until a coefficient is
128     defined.
129     \end{classdesc}
130    
131 gross 625 \begin{methoddesc}[LinearPDE]{setValue}{
132     \optional{A=Data()}\optional{, B=Data()},
133     \optional{, C=Data()}\optional{, D=Data()}
134     \optional{, X=Data()}\optional{, Y=Data()}
135     \optional{, d=Data()}\optional{, y=Data()}
136     \optional{, d_contact=Data()}\optional{, y_contact=Data()}
137     \optional{, q=Data()}\optional{, r=Data()}}
138     assigns new values to coefficients.
139     If the new coefficient value is not a \Data object, it is converted into a \Data object in the
140 jgs 102 appropriate \FunctionSpace.
141     \end{methoddesc}
142    
143     \begin{methoddesc}[LinearPDE]{getCoefficient}{name}
144     return the value assigned to coefficient \var{name}. If \var{name} is not a valid name
145     an exception is raised.
146     \end{methoddesc}
147    
148     \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name}
149     returns the shape of coefficient \var{name} even if no value has been assigned to it.
150     \end{methoddesc}
151    
152 gross 625 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name}
153 jgs 102 returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it.
154     \end{methoddesc}
155    
156     \begin{methoddesc}[LinearPDE]{setDebugOn}{}
157     switches the debug mode to on.
158     \end{methoddesc}
159    
160     \begin{methoddesc}[LinearPDE]{setDebugOff}{}
161     switches the debug mode to on.
162     \end{methoddesc}
163    
164 gross 625 \begin{methoddesc}[LinearPDE]{isUsingLumping}{}
165 gross 653 returns \True if \LUMPING is set as the solver for the system of lienar equations.
166     Otherwise \False is returned.
167 jgs 102 \end{methoddesc}
168    
169 gross 653 \begin{methoddesc}[LinearPDE]{setSolverMethod}{\optional{solver=LinearPDE.DEFAULT}\optional{, preconditioner=LinearPDE.DEFAULT}}
170 gross 625 sets the solver method and preconditioner to be used. It is pointed out that a PDE solver library
171     may not know the specified solver method but may choose a similar method and preconditioner.
172 jgs 102 \end{methoddesc}
173    
174 gross 653 \begin{methoddesc}[LinearPDE]{getSolverMethodName}{}
175     returns the name of the solver method and preconditioner which is currently been used.
176     \end{methoddesc}
177    
178     \begin{methoddesc}[LinearPDE]{getSolverMethod}{}
179     returns the solver method and preconditioner which is currently been used.
180     \end{methoddesc}
181    
182     \begin{methoddesc}[LinearPDE]{setSolverPackage}{\optional{package=LinearPDE.DEFAULT}}
183     Set the solver package to be used by PDE library to solve the linear systems of equations. The
184     specified package may not be supported by the PDE solver library. In this case, dependng on
185     the PDE solver, the default solver is used or an exeption is thrown.
186     If \var{package} is not specified, the default package of the PDE solver library is used.
187     \end{methoddesc}
188    
189     \begin{methoddesc}[LinearPDE]{getSolverPackage}{}
190     returns the linear solver package currently by the PDE solver library
191     \end{methoddesc}
192    
193    
194 gross 625 \begin{methoddesc}[LinearPDE]{setTolerance}{\optional{tol=1.e-8}}:
195     resets the tolerance for solution. The actually meaning of tolerance is
196     depending on the underlying PDE library. In most cases, the tolerance
197     will only consider the error from solving the discerete problem but will
198     not consider any discretization error.
199     \end{methoddesc}
200 jgs 102
201 gross 625 \begin{methoddesc}[LinearPDE]{getTolerance}{}
202     returns the current tolerance of the solution
203 jgs 102 \end{methoddesc}
204    
205 gross 625 \begin{methoddesc}[LinearPDE]{getDomain}{}
206     returns the \Domain of the PDE.
207 jgs 102 \end{methoddesc}
208    
209 gross 625 \begin{methoddesc}[LinearPDE]{getDim}{}
210     returns the spatial dimension of the PDE.
211 jgs 102 \end{methoddesc}
212    
213 gross 625 \begin{methoddesc}[LinearPDE]{getNumEquations}{}
214     returns the number of equations.
215     \end{methoddesc}
216 jgs 102
217 gross 625 \begin{methoddesc}[LinearPDE]{getNumSolutions}{}
218     returns the number of components of the solution.
219 jgs 102 \end{methoddesc}
220    
221 gross 625 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False}
222     returns \True if the PDE is symmetric and \False otherwise.
223     The method is very computational expensive and should only be
224     called for testing purposes. The symmetry flag is not altered.
225     If \var{verbose}=\True information about where symmetry is violated
226     are printed.
227 jgs 102 \end{methoddesc}
228    
229 gross 625 \begin{methoddesc}[LinearPDE]{getFlux}{u}
230     returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u}
231     defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively.
232 jgs 102 \end{methoddesc}
233    
234 gross 625
235 jgs 102 \begin{methoddesc}[LinearPDE]{isSymmetric}{}
236     returns \True if the PDE has been indicated to be symmetric.
237     Otherwise \False is returned.
238     \end{methoddesc}
239    
240     \begin{methoddesc}[LinearPDE]{setSymmetryOn}{}
241     indicates that the PDE is symmetric.
242     \end{methoddesc}
243    
244     \begin{methoddesc}[LinearPDE]{setSymmetryOff}{}
245     indicates that the PDE is not symmetric.
246     \end{methoddesc}
247    
248     \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{}
249 gross 625 switches on the reduction of polynomial order for the solution and equation evaluation even if
250     a quadratic or higher interpolation order is defined in the \Domain. This feature may not
251     be supported by all PDE libraries.
252 jgs 102 \end{methoddesc}
253    
254     \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{}
255     switches off the reduction of polynomial order for the solution and
256     equation evaluation.
257     \end{methoddesc}
258    
259     \begin{methoddesc}[LinearPDE]{getOperator}{}
260     returns the \Operator of the PDE.
261     \end{methoddesc}
262    
263 gross 625 \begin{methoddesc}[LinearPDE]{getRightHandSide}{}
264 jgs 102 returns the right hand side of the PDE as a \Data object. If
265     \var{ignoreConstraint}=\True the constraints are not considered
266     when building up the right hand side.
267     \end{methoddesc}
268    
269     \begin{methoddesc}[LinearPDE]{getSystem}{}
270     returns the \Operator and right hand side of the PDE.
271     \end{methoddesc}
272    
273 gross 625 \begin{methoddesc}[LinearPDE]{getSolution}{
274     \optional{verbose=False}
275     \optional{, reordering=LinearPDE.NO_REORDERING}
276     \optional{, iter_max=1000}
277     \optional{, drop_tolerance=0.01}
278     \optional{, drop_storage=1.20}
279     \optional{, truncation=-1}
280     \optional{, restart=-1}
281     }
282 gross 653 returns (an approximation of) the solution of the PDE. If \code{verbose=\True} some information during the solution process printed.
283     \var{reordering} selects a reordering methods that is applied before or during the solution process
284     (=\NOREORDERING ,\MINIMUMFILLIN ,\NESTEDDESCTION).
285     \var{iter_max} specifies the maximum number of iteration steps that are allowed to reach the specified tolerance.
286 gross 625 \var{drop_tolerance} specifies a relative tolerance for small elements to be dropped when building a preconditioner
287 gross 653 (eg. in \ILUT). \var{drop_storage} limits the extra storage allowed when building a preconditioner
288     (eg. in \ILUT). The extra storage is given relative to the size of the stiffness matrix, eg.
289 gross 625 \var{drop_storage=1.2} will allow the preconditioner to use the $1.2$ fold storage space than used
290 gross 653 for the stiffness matrix. \var{truncation} defines the truncation.
291 jgs 102 \end{methoddesc}
292    
293 gross 625 \begin{memberdesc}[LinearPDE]{DEFAULT}
294     default method, preconditioner or package to be used to solve the PDE. An appropriate method should be
295     chosen by the used PDE solver library.
296     \end{memberdesc}
297 jgs 102
298 gross 625 \begin{memberdesc}[LinearPDE]{SCSL}
299 gross 653 the SCSL library by SGI,~\Ref{SCSL}\footnote{The SCSL library will only be available on SGI systems}
300 gross 625 \end{memberdesc}
301 jgs 102
302 gross 625 \begin{memberdesc}[LinearPDE]{MKL}
303 gross 653 the MKL library by Intel,~\Ref{MKL}\footnote{The MKL library will only be available when the intel compilation environment is used.}.
304 gross 625 \end{memberdesc}
305 jgs 102
306 gross 625 \begin{memberdesc}[LinearPDE]{UMFPACK}
307 gross 653 the UMFPACK,~\Ref{UMFPACK}. Remark: UMFPACK is not parallelized.
308 gross 625 \end{memberdesc}
309 jgs 102
310 gross 625 \begin{memberdesc}[LinearPDE]{PASO}
311 gross 653 the solver library of \finley, see \Sec{CHAPTER ON FINLEY}.
312 gross 625 \end{memberdesc}
313 jgs 102
314 gross 625 \begin{memberdesc}[LinearPDE]{ITERATIVE}
315 gross 653 the default iterative method and preconditioner. The actually used method depends on the
316     PDE solver library and the solver package been choosen. Typically, \PCG is used for symmetric PDEs
317     and \BiCGStab otherwise, both with \JACOBI preconditioner.
318 gross 625 \end{memberdesc}
319 jgs 102
320 gross 625 \begin{memberdesc}[LinearPDE]{DIRECT}
321 gross 653 the default direct linear solver.
322 gross 625 \end{memberdesc}
323 jgs 102
324 gross 625 \begin{memberdesc}[LinearPDE]{CHOLEVSKY}
325 gross 653 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE.
326 gross 625 \end{memberdesc}
327 jgs 110
328 gross 625 \begin{memberdesc}[LinearPDE]{PCG}
329 gross 653 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE.
330 gross 625 \end{memberdesc}
331 jgs 110
332 gross 625 \begin{memberdesc}[LinearPDE]{GMRES}
333 gross 653 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters
334 gross 625 \var{truncation} and \var{restart} of \method{getSolution}.
335     \end{memberdesc}
336 jgs 102
337 gross 625 \begin{memberdesc}[LinearPDE]{LUMPING}
338 gross 653 uses lumping to solve the system of linear equations~\index{linear solver!lumping}\index{lumping}. This solver technique
339     condenses the stiffness matrix to a diagonal matrix so the solution of the linear systems becomes very cheap. It can be used when
340     only \var{D} is present but in any case has to applied with care. The difference in the solutions with and without lumping can be significant
341     but is expect to converge to zero when the mesh gets finer.
342     Lumping does not use the linear system solver library.
343 gross 625 \end{memberdesc}
344 jgs 107
345 gross 625 \begin{memberdesc}[LinearPDE]{PRES20}
346 gross 653 the GMRES method with truncation after five residuals and
347 gross 625 restart after 20 steps, see~\Ref{WEISS}.
348 gross 653 \end{memberdesc}[LinearPDE]{CR}
349 gross 625
350     \begin{memberdesc}[LinearPDE]{CGS}
351     conjugate gradient squared method, see~\Ref{WEISS}.
352 jgs 107 \end{memberdesc}
353    
354 gross 625 \begin{memberdesc}[LinearPDE]{BICGSTAB}
355 gross 653 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}.
356 jgs 107 \end{memberdesc}
357    
358 gross 625 \begin{memberdesc}[LinearPDE]{SSOR}
359 gross 653 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support
360     this as a solver.
361 gross 625 \end{memberdesc}
362     \begin{memberdesc}[LinearPDE]{ILU0}
363 gross 653 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}.
364     \end{memberdesc}
365    
366 gross 625 \begin{memberdesc}[LinearPDE]{ILUT}
367 gross 653 the incomplete LU factorization preconditioner with fill-in, see~\Ref{Saad}. During the LU-factorization element with
368     relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the
369     \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the
370     \method{getSolution} call.
371     \end{memberdesc}
372    
373 gross 625 \begin{memberdesc}[LinearPDE]{JACOBI}
374 gross 653 the Jacobi preconditioner, see~\Ref{Saad}.
375     \end{memberdesc}
376    
377 gross 625 \begin{memberdesc}[LinearPDE]{AMG}
378 gross 653 the algebraic--multi grid method, see~\Ref{AMG}. This method can be used as linear solver method but is more robust when used
379     in a preconditioner.
380     \end{memberdesc}
381    
382 gross 625 \begin{memberdesc}[LinearPDE]{RILU}
383 gross 653 recursive incomplete LU factorization preconditioner, see~\Ref{RILU}. This method is similar to \ILUT but uses smoothing
384     between levels. During the LU-factorization element with
385     relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the
386     \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the
387     \method{getSolution} call.
388     \end{memberdesc}
389 jgs 107
390 gross 653 \begin{memberdesc}[LinearPDE]{NO_REORDERING}
391     no ordering is used during factorization.
392     \end{memberdesc}
393 gross 625
394 gross 653 \begin{memberdesc}[LinearPDE]{MINIMUM_FILL_IN}
395     applies reordering before factorization using a fill-in minimization strategy. You have to check with the particular solver library or
396     linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in.
397     \end{memberdesc}
398 gross 625
399     \begin{memberdesc}[LinearPDE]{NESTED_DISSECTION}
400 gross 653 applies reordering before factorization using a nested dissection strategy. You have to check with the particular solver library or
401     linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in.
402     \end{memberdesc}
403 gross 625
404 jgs 102 \section{The \Poisson Class}
405     The \Poisson class provides an easy way to define and solve the Poisson
406     equation
407     \begin{equation}\label{POISSON.1}
408     -u\hackscore{,ii}=f\; .
409     \end{equation}
410     with homogeneous boundary conditions
411     \begin{equation}\label{POISSON.2}
412     n\hackscore{i}u\hackscore{,i}=0
413     \end{equation}
414     and homogeneous constraints
415     \begin{equation}\label{POISSON.3}
416     u=0 \mbox{ where } q>0
417     \end{equation}
418     $f$ has to be a \Scalar in the \Function and $q$ must be
419     a \Scalar in the \SolutionFS.
420    
421     \begin{classdesc}{Poisson}{domain}
422     opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE.
423     \end{classdesc}
424     \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()}
425     assigns new values to \var{f} and \var{q}.
426     \end{methoddesc}
427 gross 625
428     \section{The \Helmholtz Class}
429    
430     \section{The \Lame Class}
431    

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