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more work on the dary solver 

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3 %
4 % Copyright (c) 2003-2008 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
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8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
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15 \chapter{The Module \linearPDEs}
19 \section{Linear Partial Differential Equations}
20 \label{SEC LinearPDE}
22 The \LinearPDE class is used to define a general linear, steady, second order PDE
23 for an unknown function $u$ on a given $\Omega$ defined through a \Domain object.
24 In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes
25 the outer normal field on $\Gamma$.
27 For a single PDE with a solution with a single component the linear PDE is defined in the
28 following form:
29 \begin{equation}\label{LINEARPDE.SINGLE.1}
30 -(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 \; .
31 \end{equation}
32 $u_{,j}$ 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.
33 The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the
34 \Function on the PDE or objects that can be converted into such \Data objects.
35 $A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar.
36 The following natural
37 boundary conditions are considered \index{boundary condition!natural} on $\Gamma$:
38 \begin{equation}\label{LINEARPDE.SINGLE.2}
39 n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;.
40 \end{equation}
41 Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are
42 each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the
43 solution at certain locations in the domain. They have the form
44 \begin{equation}\label{LINEARPDE.SINGLE.3}
45 u=r \mbox{ where } q>0
46 \end{equation}
47 $r$ and $q$ are each \Scalar where $q$ is the characteristic function
48 \index{characteristic function} defining where the constraint is applied.
49 The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1}
50 or \eqn{LINEARPDE.SINGLE.2}.
52 For a system of PDEs and a solution with several components the PDE has the form
53 \begin{equation}\label{LINEARPDE.SYSTEM.1}
54 -(A\hackscore{ijkl} u\hackscore{k,l})\hackscore{,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} \; .
55 \end{equation}
56 $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne.
57 The natural boundary conditions \index{boundary condition!natural} take the form:
58 \begin{equation}\label{LINEARPDE.SYSTEM.2}
59 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} \;.
60 \end{equation}
61 The coefficient $d$ is a \RankTwo and $y$ is a
62 \RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form
63 \begin{equation}\label{LINEARPDE.SYSTEM.3}
64 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0
65 \end{equation}
66 $r$ and $q$ are each \RankOne. Notice that not necessarily all components must
67 have a constraint at all locations.
69 \LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$
70 in the domain $\Omega$. To specify the conditions across the discontinuity we are using the
71 generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution
72 defined as
73 \begin{equation}\label{LINEARPDE.SYSTEM.5}
74 J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij}
75 \end{equation}
76 For the case of single solution component and single PDE $J$ is defined
77 \begin{equation}\label{LINEARPDE.SINGLE.5}
78 J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j}
79 \end{equation}
80 In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the
81 discontinuity pointing from side 0 towards side 1. For a system of PDEs
82 the contact condition takes the form
83 \begin{equation}\label{LINEARPDE.SYSTEM.6}
84 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} \; .
85 \end{equation}
86 where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the
87 discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference
88 of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$.
89 The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a
90 \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne.
91 In case of a single PDE and a single component solution the contact condition takes the form
92 \begin{equation}\label{LINEARPDE.SINGLE.6}
93 n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u]
94 \end{equation}
95 In this case the the coefficient $d^{contact}$ and $y^{contact}$ are each \Scalar
96 both in the \FunctionOnContactZero or \FunctionOnContactOne.
98 The PDE is symmetrical \index{symmetrical} if
99 \begin{equation}\label{LINEARPDE.SINGLE.4}
100 A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j}
101 \end{equation}
102 The system of PDEs is symmetrical \index{symmetrical} if
103 \begin{eqnarray}
104 \label{LINEARPDE.SYSTEM.4}
105 A\hackscore{ijkl}&=&A\hackscore{klij} \\
106 B\hackscore{ijk}&=&C\hackscore{kij} \\
107 D\hackscore{ik}&=&D\hackscore{ki} \\
108 d\hackscore{ik}&=&d\hackscore{ki} \\
109 d^{contact}\hackscore{ik}&=&d^{contact}\hackscore{ki}
110 \end{eqnarray}
111 Note that in contrast with the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$
112 have to be inspected.
115 \subsection{Classes}
116 \declaremodule{extension}{esys.escript.linearPDEs}
117 \modulesynopsis{Linear partial differential equation handler}
118 The module \linearPDEs provides an interface to define and solve linear partial
119 differential equations within \escript. The module \linearPDEs does not provide any
120 solver capabilities in itself but hands the PDE over to
121 the PDE solver library defined through the \Domain of the PDE.
122 The general interface is provided through the \LinearPDE class. The
123 \AdvectivePDE which is derived from the \LinearPDE class
124 provides an interface to a PDE dominated by its advective terms. The \Poisson
125 class which is also derived form the \LinearPDE class should be used
126 to define the Poisson equation \index{Poisson}.
128 \subsection{\LinearPDE class}
129 This is the general class to define a linear PDE in \escript. We list a selection of the most
130 important methods of the class. For a complete list, see the reference at \ReferenceGuide.
132 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0}
133 opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations}
134 and \var{numSolutions} gives the number of equations and the number of solution components.
135 If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations
136 and the number solutions, respectively, stay undefined until a coefficient is
137 defined.
138 \end{classdesc}
140 \subsubsection{\LinearPDE methods}
142 \begin{methoddesc}[LinearPDE]{setValue}{
143 \optional{A}\optional{, B},
144 \optional{, C}\optional{, D}
145 \optional{, X}\optional{, Y}
146 \optional{, d}\optional{, y}
147 \optional{, d_contact}\optional{, y_contact}
148 \optional{, q}\optional{, r}}
149 assigns new values to coefficients. By default all values are assumed to be zero\footnote{
150 In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The
151 can by used by the solver library to reduce computational costs.
152 }
153 If the new coefficient value is not a \Data object, it is converted into a \Data object in the
154 appropriate \FunctionSpace.
155 \end{methoddesc}
157 \begin{methoddesc}[LinearPDE]{getCoefficient}{name}
158 return the value assigned to coefficient \var{name}. If \var{name} is not a valid name
159 an exception is raised.
160 \end{methoddesc}
162 \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name}
163 returns the shape of coefficient \var{name} even if no value has been assigned to it.
164 \end{methoddesc}
166 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name}
167 returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it.
168 \end{methoddesc}
170 \begin{methoddesc}[LinearPDE]{setDebugOn}{}
171 switches on debug mode.
172 \end{methoddesc}
174 \begin{methoddesc}[LinearPDE]{setDebugOff}{}
175 switches off debug mode.
176 \end{methoddesc}
178 \begin{methoddesc}[LinearPDE]{isUsingLumping}{}
179 returns \True if \LUMPING is set as the solver for the system of linear equations.
180 Otherwise \False is returned.
181 \end{methoddesc}
183 \begin{methoddesc}[LinearPDE]{setSolverMethod}{\optional{solver=LinearPDE.DEFAULT}\optional{, preconditioner=LinearPDE.DEFAULT}}
184 sets the solver method and preconditioner to be used. It should be noted that a PDE solver library
185 may not know the specified solver method but may choose a similar method and preconditioner.
186 \end{methoddesc}
188 \begin{methoddesc}[LinearPDE]{getSolverMethodName}{}
189 returns the name of the solver method and preconditioner which is in use.
190 \end{methoddesc}
192 \begin{methoddesc}[LinearPDE]{getSolverMethod}{}
193 returns the solver method and preconditioner which is in use.
194 \end{methoddesc}
196 \begin{methoddesc}[LinearPDE]{setSolverPackage}{\optional{package=LinearPDE.DEFAULT}}
197 sets the solver package to be used by PDE library to solve the linear systems of equations. The
198 specified package may not be supported by the PDE solver library. In this case, depending on
199 the PDE solver, the default solver is used or an exception is thrown.
200 If \var{package} is not specified, the default package of the PDE solver library is used.
201 \end{methoddesc}
203 \begin{methoddesc}[LinearPDE]{getSolverPackage}{}
204 returns the linear solver package currently by the PDE solver library
205 \end{methoddesc}
208 \begin{methoddesc}[LinearPDE]{setTolerance}{\optional{tol=1.e-8}}
209 resets the tolerance for solution. The actually meaning of tolerance depends
210 on the underlying PDE library. In most cases, the tolerance
211 will only consider the error from solving the discrete problem but will
212 not consider any discretization error.
213 \end{methoddesc}
215 \begin{methoddesc}[LinearPDE]{setToleranceReductionFactor}{TOL}
216 lowers the tolerance by a factor of TOL.
217 \end{methoddesc}
219 \begin{methoddesc}[LinearPDE]{getTolerance}{}
220 returns the current tolerance of the solution
221 \end{methoddesc}
223 \begin{methoddesc}[LinearPDE]{getDomain}{}
224 returns the \Domain of the PDE.
225 \end{methoddesc}
227 \begin{methoddesc}[LinearPDE]{getDim}{}
228 returns the spatial dimension of the PDE.
229 \end{methoddesc}
231 \begin{methoddesc}[LinearPDE]{getNumEquations}{}
232 returns the number of equations.
233 \end{methoddesc}
235 \begin{methoddesc}[LinearPDE]{getNumSolutions}{}
236 returns the number of components of the solution.
237 \end{methoddesc}
239 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False}
240 returns \True if the PDE is symmetric and \False otherwise.
241 The method is very computationally expensive and should only be
242 called for testing purposes. The symmetry flag is not altered.
243 If \var{verbose}=\True information about where symmetry is violated
244 are printed.
245 \end{methoddesc}
247 \begin{methoddesc}[LinearPDE]{getFlux}{u}
248 returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u}
249 defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively.
250 \end{methoddesc}
253 \begin{methoddesc}[LinearPDE]{isSymmetric}{}
254 returns \True if the PDE has been indicated to be symmetric.
255 Otherwise \False is returned.
256 \end{methoddesc}
258 \begin{methoddesc}[LinearPDE]{setSymmetryOn}{}
259 indicates that the PDE is symmetric.
260 \end{methoddesc}
262 \begin{methoddesc}[LinearPDE]{setSymmetryOff}{}
263 indicates that the PDE is not symmetric.
264 \end{methoddesc}
266 \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{}
267 switches on the reduction of polynomial order for the solution and equation evaluation even if
268 a quadratic or higher interpolation order is defined in the \Domain. This feature may not
269 be supported by all PDE libraries.
270 \end{methoddesc}
272 \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{}
273 switches off the reduction of polynomial order for the solution and
274 equation evaluation.
275 \end{methoddesc}
277 \begin{methoddesc}[LinearPDE]{getOperator}{}
278 returns the \Operator of the PDE.
279 \end{methoddesc}
281 \begin{methoddesc}[LinearPDE]{getRightHandSide}{}
282 returns the right hand side of the PDE as a \Data object. If
283 \var{ignoreConstraint}=\True, then the constraints are not considered
284 when building up the right hand side.
285 \end{methoddesc}
287 \begin{methoddesc}[LinearPDE]{getSystem}{}
288 returns the \Operator and right hand side of the PDE.
289 \end{methoddesc}
291 \begin{methoddesc}[LinearPDE]{getSolution}{
292 \optional{verbose=False}
293 \optional{, reordering=LinearPDE.NO_REORDERING}
294 \optional{, iter_max=1000}
295 \optional{, drop_tolerance=0.01}
296 \optional{, drop_storage=1.20}
297 \optional{, truncation=-1}
298 \optional{, restart=-1}
299 }
300 returns (an approximation of) the solution of the PDE. If \code{verbose=\True}, then some information is printed during the solution process.
301 \var{reordering} selects a reordering methods that is applied before or during the solution process
303 \var{iter_max} specifies the maximum number of iteration steps that are allowed to reach the specified tolerance.
304 \var{drop_tolerance} specifies a relative tolerance for small elements to be dropped when building a preconditioner
305 (eg. in \ILUT). \var{drop_storage} limits the extra storage allowed when building a preconditioner
306 (eg. in \ILUT). The extra storage is given relative to the size of the stiffness matrix, eg.
307 \var{drop_storage=1.2} will allow the preconditioner to use the $1.2$ fold storage space than used
308 for the stiffness matrix. \var{truncation} defines the truncation.
309 \end{methoddesc}
311 \subsubsection{\LinearPDE symbols/members}
313 \begin{memberdesc}[LinearPDE]{DEFAULT}
314 default method, preconditioner or package to be used to solve the PDE. An appropriate method should be
315 chosen by the used PDE solver library.
316 \end{memberdesc}
318 \begin{memberdesc}[LinearPDE]{SCSL}
319 the SCSL library by SGI,~\Ref{SCSL}\footnote{The SCSL library will only be available on SGI systems}
320 \end{memberdesc}
322 \begin{memberdesc}[LinearPDE]{MKL}
323 the MKL library by Intel,~\Ref{MKL}\footnote{The MKL library will only be available when the Intel compilation environment is used.}.
324 \end{memberdesc}
326 \begin{memberdesc}[LinearPDE]{UMFPACK}
327 the UMFPACK,~\Ref{UMFPACK}. Remark: UMFPACK is not parallelized.
328 \end{memberdesc}
330 \begin{memberdesc}[LinearPDE]{PASO}
331 the solver library of \finley, see \Sec{CHAPTER ON FINLEY}.
332 \end{memberdesc}
334 \begin{memberdesc}[LinearPDE]{ITERATIVE}
335 the default iterative method and preconditioner. The actually used method depends on the
336 PDE solver library and the solver package been chosen. Typically, \PCG is used for symmetric PDEs
337 and \BiCGStab otherwise, both with \JACOBI preconditioner.
338 \end{memberdesc}
340 \begin{memberdesc}[LinearPDE]{DIRECT}
341 the default direct linear solver.
342 \end{memberdesc}
344 \begin{memberdesc}[LinearPDE]{CHOLEVSKY}
345 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE.
346 \end{memberdesc}
348 \begin{memberdesc}[LinearPDE]{PCG}
349 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE.
350 \end{memberdesc}
352 \begin{memberdesc}[LinearPDE]{TFQMR}
353 transpose-free quasi-minimal residual method, see~\Ref{WEISS}\index{linear solver!TFQMR}\index{TFQMR}. \end{memberdesc}
355 \begin{memberdesc}[LinearPDE]{GMRES}
356 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters
357 \var{truncation} and \var{restart} of \method{getSolution}.
358 \end{memberdesc}
360 \begin{memberdesc}[LinearPDE]{MINRES}
361 minimal residual method method, \index{linear solver!MINRES}\index{MINRES} \end{memberdesc}
363 \begin{memberdesc}[LinearPDE]{LUMPING}
364 uses lumping to solve the system of linear equations~\index{linear solver!lumping}\index{lumping}. This solver technique
365 condenses the stiffness matrix to a diagonal matrix so the solution of the linear systems becomes very cheap. It can be used when
366 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
367 but is expected to converge to zero when the mesh gets finer.
368 Lumping does not use the linear system solver library.
369 \end{memberdesc}
371 \begin{memberdesc}[LinearPDE]{PRES20}
372 the GMRES method with truncation after five residuals and
373 restart after 20 steps, see~\Ref{WEISS}.
374 \end{memberdesc}
376 \begin{memberdesc}[LinearPDE]{CGS}
377 conjugate gradient squared method, see~\Ref{WEISS}.
378 \end{memberdesc}
380 \begin{memberdesc}[LinearPDE]{BICGSTAB}
381 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}.
382 \end{memberdesc}
384 \begin{memberdesc}[LinearPDE]{SSOR}
385 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support
386 this as a solver.
387 \end{memberdesc}
388 \begin{memberdesc}[LinearPDE]{ILU0}
389 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}.
390 \end{memberdesc}
392 \begin{memberdesc}[LinearPDE]{ILUT}
393 the incomplete LU factorization preconditioner with fill-in, see~\Ref{Saad}. During the LU-factorization element with
394 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the
395 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the
396 \method{getSolution} call.
397 \end{memberdesc}
399 \begin{memberdesc}[LinearPDE]{JACOBI}
400 the Jacobi preconditioner, see~\Ref{Saad}.
401 \end{memberdesc}
403 \begin{memberdesc}[LinearPDE]{AMG}
404 the algebraic--multi grid method, see~\Ref{AMG}. This method can be used as linear solver method but is more robust when used
405 in a preconditioner.
406 \end{memberdesc}
408 \begin{memberdesc}[LinearPDE]{GS}
409 the symmetric Gauss-Seidel preconditioner, see~\Ref{Saad}.
410 \end{memberdesc}
412 \begin{memberdesc}[LinearPDE]{RILU}
413 recursive incomplete LU factorization preconditioner, see~\Ref{RILU}. This method is similar to \ILUT but uses smoothing
414 between levels. During the LU-factorization element with
415 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the
416 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the
417 \method{getSolution} call.
418 \end{memberdesc}
420 \begin{memberdesc}[LinearPDE]{NO_REORDERING}
421 no ordering is used during factorization.
422 \end{memberdesc}
424 \begin{memberdesc}[LinearPDE]{MINIMUM_FILL_IN}
425 applies reordering before factorization using a fill-in minimization strategy. You have to check with the particular solver library or
426 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in.
427 \end{memberdesc}
429 \begin{memberdesc}[LinearPDE]{NESTED_DISSECTION}
430 applies reordering before factorization using a nested dissection strategy. You have to check with the particular solver library or
431 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in.
432 \end{memberdesc}
434 \subsection{The \Poisson Class}
435 The \Poisson class provides an easy way to define and solve the Poisson
436 equation
437 \begin{equation}\label{POISSON.1}
438 -u\hackscore{,ii}=f\; .
439 \end{equation}
440 with homogeneous boundary conditions
441 \begin{equation}\label{POISSON.2}
442 n\hackscore{i}u\hackscore{,i}=0
443 \end{equation}
444 and homogeneous constraints
445 \begin{equation}\label{POISSON.3}
446 u=0 \mbox{ where } q>0
447 \end{equation}
448 $f$ has to be a \Scalar in the \Function and $q$ must be
449 a \Scalar in the \SolutionFS.
451 \begin{classdesc}{Poisson}{domain}
452 opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE.
453 \end{classdesc}
454 \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()}
455 assigns new values to \var{f} and \var{q}.
456 \end{methoddesc}
458 \subsection{The \Helmholtz Class}
459 The \Helmholtz class defines the Helmholtz problem
460 \begin{equation}\label{HZ.1}
461 \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f
462 \end{equation}
463 with natural boundary conditions
464 \begin{equation}\label{HZ.2}
465 k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u
466 \end{equation}
467 and constraints:
468 \begin{equation}\label{HZ.3}
469 u=r \mbox{ where } q>0
470 \end{equation}
471 $\omega$, $k$, $f$ have to be a \Scalar in the \Function,
472 $g$ and $\alpha$ must be a \Scalar in the \FunctionOnBoundary,
473 and $q$ and $r$ must be a \Scalar in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace.
475 \begin{classdesc}{Helmholtz}{domain}
476 opens a Helmholtz equation on the \Domain domain. \Helmholtz is derived from \LinearPDE.
477 \end{classdesc}
478 \begin{methoddesc}[Helmholtz]{setValue}{ \optional{omega} \optional{, k} \optional{, f} \optional{, alpha} \optional{, g} \optional{, r} \optional{, q}}
479 assigns new values to \var{omega}, \var{k}, \var{f}, \var{alpha}, \var{g}, \var{r}, \var{q}. By default all values are set to be zero.
480 \end{methoddesc}
482 \subsection{The \Lame Class}
483 The \Lame class defines a Lame equation problem:
484 \begin{equation}\label{LE.1}
485 -\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j}
486 \end{equation}
487 with natural boundary conditions:
488 \begin{equation}\label{LE.2}
489 n\hackscore{j}(\mu \; (u\hackscore{i,j}+u\hackscore{j,i})+\lambda*u\hackscore{k,k}) = f\hackscore{i}+n\hackscore{j}\sigma\hackscore{ij}
490 \end{equation}
491 and constraint
492 \begin{equation}\label{LE.3}
493 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0
494 \end{equation}
495 $\mu$, $\lambda$ have to be a \Scalar in the \Function,
496 $F$ has to be a \Vector in the \Function,
497 $\sigma$ has to be a \Tensor in the \Function,
498 $f$ must be a \Vector in the \FunctionOnBoundary,
499 and $q$ and $r$ must be a \Vector in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace.
501 \begin{classdesc}{Lame}{domain}
502 opens a Lame equation on the \Domain domain. \Lame is derived from \LinearPDE.
503 \end{classdesc}
504 \begin{methoddesc}[Lame]{setValue}{ \optional{lame_lambda} \optional{, lame_mu} \optional{, F} \optional{, sigma} \optional{, f} \optional{, r} \optional{, q}}
505 assigns new values to
506 \var{lame_lambda},
507 \var{lame_mu},
508 \var{F},
509 \var{sigma},
510 \var{f},
511 \var{r} and
512 \var{q}
513 By default all values are set to be zero.
514 \end{methoddesc}
516 % \section{Transport Problems}
517 % \label{SEC Transport}


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