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1 % $Id$
2 %
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
10
11 \chapter{The module \linearPDEs}
12
13 \declaremodule{extension}{linearPDEs} \modulesynopsis{Linear partial pifferential equation handler}
14 The module \linearPDEs provides an interface to define and solve linear partial
15 differential equations within \escript. \linearPDEs does not provide any
16 solver capabilities in itself but hands the PDE over to
17 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 For a single PDE with a solution with a single component the linear PDE is defined in the
33 following form:
34 \begin{equation}\label{LINEARPDE.SINGLE.1}
35 -(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 \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 or \eqn{LINEARPDE.SINGLE.2}.
56
57 For a system of PDEs and a solution with several components the PDE has the form
58 \begin{equation}\label{LINEARPDE.SYSTEM.1}
59 -(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 \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 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 \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 $r$ and $q$ are each \RankOne. Notice that not necessarily all components must
72 have a constraint at all locations.
73
74 \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
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 \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 \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 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 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name}
153 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 \begin{methoddesc}[LinearPDE]{isUsingLumping}{}
165 returns \True if \LUMPING is set as the solver for the system of lienar equations.
166 Otherwise \False is returned.
167 \end{methoddesc}
168
169 \begin{methoddesc}[LinearPDE]{setSolverMethod}{\optional{solver=LinearPDE.DEFAULT}\optional{, preconditioner=LinearPDE.DEFAULT}}
170 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 \end{methoddesc}
173
174 \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 \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
201 \begin{methoddesc}[LinearPDE]{getTolerance}{}
202 returns the current tolerance of the solution
203 \end{methoddesc}
204
205 \begin{methoddesc}[LinearPDE]{getDomain}{}
206 returns the \Domain of the PDE.
207 \end{methoddesc}
208
209 \begin{methoddesc}[LinearPDE]{getDim}{}
210 returns the spatial dimension of the PDE.
211 \end{methoddesc}
212
213 \begin{methoddesc}[LinearPDE]{getNumEquations}{}
214 returns the number of equations.
215 \end{methoddesc}
216
217 \begin{methoddesc}[LinearPDE]{getNumSolutions}{}
218 returns the number of components of the solution.
219 \end{methoddesc}
220
221 \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 \end{methoddesc}
228
229 \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 \end{methoddesc}
233
234
235 \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 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 \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 \begin{methoddesc}[LinearPDE]{getRightHandSide}{}
264 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 \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 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 \var{drop_tolerance} specifies a relative tolerance for small elements to be dropped when building a preconditioner
287 (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 \var{drop_storage=1.2} will allow the preconditioner to use the $1.2$ fold storage space than used
290 for the stiffness matrix. \var{truncation} defines the truncation.
291 \end{methoddesc}
292
293 \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
298 \begin{memberdesc}[LinearPDE]{SCSL}
299 the SCSL library by SGI,~\Ref{SCSL}\footnote{The SCSL library will only be available on SGI systems}
300 \end{memberdesc}
301
302 \begin{memberdesc}[LinearPDE]{MKL}
303 the MKL library by Intel,~\Ref{MKL}\footnote{The MKL library will only be available when the intel compilation environment is used.}.
304 \end{memberdesc}
305
306 \begin{memberdesc}[LinearPDE]{UMFPACK}
307 the UMFPACK,~\Ref{UMFPACK}. Remark: UMFPACK is not parallelized.
308 \end{memberdesc}
309
310 \begin{memberdesc}[LinearPDE]{PASO}
311 the solver library of \finley, see \Sec{CHAPTER ON FINLEY}.
312 \end{memberdesc}
313
314 \begin{memberdesc}[LinearPDE]{ITERATIVE}
315 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 \end{memberdesc}
319
320 \begin{memberdesc}[LinearPDE]{DIRECT}
321 the default direct linear solver.
322 \end{memberdesc}
323
324 \begin{memberdesc}[LinearPDE]{CHOLEVSKY}
325 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE.
326 \end{memberdesc}
327
328 \begin{memberdesc}[LinearPDE]{PCG}
329 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE.
330 \end{memberdesc}
331
332 \begin{memberdesc}[LinearPDE]{GMRES}
333 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters
334 \var{truncation} and \var{restart} of \method{getSolution}.
335 \end{memberdesc}
336
337 \begin{memberdesc}[LinearPDE]{LUMPING}
338 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 \end{memberdesc}
344
345 \begin{memberdesc}[LinearPDE]{PRES20}
346 the GMRES method with truncation after five residuals and
347 restart after 20 steps, see~\Ref{WEISS}.
348 \end{memberdesc}[LinearPDE]{CR}
349
350 \begin{memberdesc}[LinearPDE]{CGS}
351 conjugate gradient squared method, see~\Ref{WEISS}.
352 \end{memberdesc}
353
354 \begin{memberdesc}[LinearPDE]{BICGSTAB}
355 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}.
356 \end{memberdesc}
357
358 \begin{memberdesc}[LinearPDE]{SSOR}
359 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support
360 this as a solver.
361 \end{memberdesc}
362 \begin{memberdesc}[LinearPDE]{ILU0}
363 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}.
364 \end{memberdesc}
365
366 \begin{memberdesc}[LinearPDE]{ILUT}
367 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 \begin{memberdesc}[LinearPDE]{JACOBI}
374 the Jacobi preconditioner, see~\Ref{Saad}.
375 \end{memberdesc}
376
377 \begin{memberdesc}[LinearPDE]{AMG}
378 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 \begin{memberdesc}[LinearPDE]{RILU}
383 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
390 \begin{memberdesc}[LinearPDE]{NO_REORDERING}
391 no ordering is used during factorization.
392 \end{memberdesc}
393
394 \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
399 \begin{memberdesc}[LinearPDE]{NESTED_DISSECTION}
400 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
404 \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
428 \section{The \Helmholtz Class}
429
430 \section{The \Lame Class}
431

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