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Lame equation is corrected. Options for using smoother in AMG is added. A book on AMG is cited for standard coarsening. 
1
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3 %
4 % Copyright (c) 2003-2010 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
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12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
13
14
15 \chapter{The Module \linearPDEs}
16
17
18
19 \section{Linear Partial Differential Equations}
20 \label{SEC LinearPDE}
21
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 $\Gamma$.
26
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}.
51
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.
68
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 $J$\footnote{In some applications the definition of flux used here can be different from the commonly used definition. For instance, if $T$ is a temperature field the heat flux $q$ is defined as $q\hackscore{,i}=-\kappa T\hackscore{,i}$ ($\kappa$ is diffusifity) which differs from the definition used here by the sign. This needs to be kept in mind when defining natural boundary conditions.\index{boundary condition!natural}} 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.
97
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.
113
114 The following example illustrates the typical usage of the \LinearPDE class:
115 \begin{python}
116 from esys.escript import *
117 from esys.escript.linearPDEs import LinearPDE
118 from esys.finley import Rectangle
119 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
120 mypde=LinearPDE(mydomain)
121 mypde.setSymmetryOn()
122 mypde.setValue(A=kappa*kronecker(mydomain),D=1,Y=1)
123 u=mypde.getSolution()
124 \end{python}
125 We refer to chapter~\ref{CHAP: Tutorial} for more details.
126
127 An instance of the \SolverOptions class is attached to the \LinearPDE class object. It is used to set options of the solver used to solve the PDE. In the following
128 code the \method{getSolverOptions} is used to access the \SolverOptions
129 attached to \var{mypde}:
130 \begin{python}
131 from esys.escript import *
132 from esys.escript.linearPDEs import LinearPDE, SolverOptions
133 from esys.finley import Rectangle
134 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20)
135 mypde=LinearPDE(mydomain)
136 mypde.setValue(A=kappa*kronecker(mydomain),D=1,Y=1)
137 mypde.getSolverOptions().setVerbosityOn()
138 mypde.getSolverOptions().setSolverMethod(SolverOptions.PCG)
139 mypde.getSolverOptions().setPreconditioner(SolverOptions.AMG)
140 mypde.getSolverOptions().setTolerance(1e-8)
141 mypde.getSolverOptions().setIterMax(1000)
142 u=mypde.getSolution()
143 \end{python}
144 In this code the preconditioned conjugate gradient method \PCG
145 with preconditioner \AMG. The relative tolerance is set to $10^{-8}$ and
146 the maximum number of iteration steps to $1000$.
147
148 Moreover, after a completed solution call
149 the attached \SolverOptions object gives access to diagnostic informations:
150 \begin{python}
151 u=mypde.getSolution()
152 print 'Number of iteration steps =', mypde.getDiagnostics('num_iter')
153 print 'Total solution time =', mypde.getDiagnostics('time')
154 print 'Set-up time =', mypde.getDiagnostics('set_up_time')
155 print 'Net time =', mypde.getDiagnostics('net_time')
156 print 'Residual norm of returned solution =', mypde.getDiagnostics('residual_norm')
157 \end{python}
158 Typically a negative value for a diagnostic value indicates that the value is undefined.
159
160 \subsection{Classes}
161 \declaremodule{extension}{esys.escript.linearPDEs}
162 \modulesynopsis{Linear partial differential equation handler}
163 The module \linearPDEs provides an interface to define and solve linear partial
164 differential equations within \escript. The module \linearPDEs does not provide any
165 solver capabilities in itself but hands the PDE over to
166 the PDE solver library defined through the \Domain of the PDE, eg. \finley.
167 The general interface is provided through the \LinearPDE class. The \Poisson
168 class which is also derived form the \LinearPDE class should be used
169 to define the Poisson equation \index{Poisson}.
170
171 \subsection{\LinearPDE class}
172 This is the general class to define a linear PDE in \escript. We list a selection of the most
173 important methods of the class. For a complete list, see the reference at \ReferenceGuide.
174
175 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0}
176 opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations}
177 and \var{numSolutions} gives the number of equations and the number of solution components.
178 If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations
179 and the number solutions, respectively, stay undefined until a coefficient is
180 defined.
181 \end{classdesc}
182
183 \subsubsection{\LinearPDE methods}
184
185 \begin{methoddesc}[LinearPDE]{setValue}{
186 \optional{A}\optional{, B},
187 \optional{, C}\optional{, D}
188 \optional{, X}\optional{, Y}
189 \optional{, d}\optional{, y}
190 \optional{, d_contact}\optional{, y_contact}
191 \optional{, q}\optional{, r}}
192 assigns new values to coefficients. By default all values are assumed to be zero\footnote{
193 In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The
194 can by used by the solver library to reduce computational costs.
195 }
196 If the new coefficient value is not a \Data object, it is converted into a \Data object in the
197 appropriate \FunctionSpace.
198 \end{methoddesc}
199
200 \begin{methoddesc}[LinearPDE]{getCoefficient}{name}
201 return the value assigned to coefficient \var{name}. If \var{name} is not a valid name
202 an exception is raised.
203 \end{methoddesc}
204
205 \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name}
206 returns the shape of coefficient \var{name} even if no value has been assigned to it.
207 \end{methoddesc}
208
209 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name}
210 returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it.
211 \end{methoddesc}
212
213 \begin{methoddesc}[LinearPDE]{setDebugOn}{}
214 switches on debug mode.
215 \end{methoddesc}
216
217 \begin{methoddesc}[LinearPDE]{setDebugOff}{}
218 switches off debug mode.
219 \end{methoddesc}
220
221 \begin{methoddesc}[LinearPDE]{getSolverOptions}{}
222 returns the solver options for solving the PDE. In fact the method returns
223 a \SolverOptions class object which can be used to modify the tolerance,
224 the solver or the preconditioner, see Section~\ref{SEC Solver Options} for details.
225 \end{methoddesc}
226
227 \begin{methoddesc}[LinearPDE]{setSolverOptions}{\optional{options=None}}
228 sets the solver options for solving the PDE. If argument \var{options} is present it
229 must be a \SolverOptions class object, see Section~\ref{SEC Solver Options} for details. Otherwise the solver options are reset to the default.
230 \end{methoddesc}
231
232
233 \begin{methoddesc}[LinearPDE]{isUsingLumping}{}
234 returns \True if \LUMPING is set as the solver for the system of linear equations.
235 Otherwise \False is returned.
236 \end{methoddesc}
237
238
239 \begin{methoddesc}[LinearPDE]{getDomain}{}
240 returns the \Domain of the PDE.
241 \end{methoddesc}
242
243 \begin{methoddesc}[LinearPDE]{getDim}{}
244 returns the spatial dimension of the PDE.
245 \end{methoddesc}
246
247 \begin{methoddesc}[LinearPDE]{getNumEquations}{}
248 returns the number of equations.
249 \end{methoddesc}
250
251 \begin{methoddesc}[LinearPDE]{getNumSolutions}{}
252 returns the number of components of the solution.
253 \end{methoddesc}
254
255 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False}
256 returns \True if the PDE is symmetric and \False otherwise.
257 The method is very computationally expensive and should only be
258 called for testing purposes. The symmetry flag is not altered.
259 If \var{verbose}=\True information about where symmetry is violated
260 are printed.
261 \end{methoddesc}
262
263 \begin{methoddesc}[LinearPDE]{getFlux}{u}
264 returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u}
265 defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively.
266 \end{methoddesc}
267
268
269 \begin{methoddesc}[LinearPDE]{isSymmetric}{}
270 returns \True if the PDE has been indicated to be symmetric.
271 Otherwise \False is returned.
272 \end{methoddesc}
273
274 \begin{methoddesc}[LinearPDE]{setSymmetryOn}{}
275 indicates that the PDE is symmetric.
276 \end{methoddesc}
277
278 \begin{methoddesc}[LinearPDE]{setSymmetryOff}{}
279 indicates that the PDE is not symmetric.
280 \end{methoddesc}
281
282 \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{}
283 switches on the reduction of polynomial order for the solution and equation evaluation even if
284 a quadratic or higher interpolation order is defined in the \Domain. This feature may not
285 be supported by all PDE libraries.
286 \end{methoddesc}
287
288 \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{}
289 switches off the reduction of polynomial order for the solution and
290 equation evaluation.
291 \end{methoddesc}
292
293 \begin{methoddesc}[LinearPDE]{getOperator}{}
294 returns the \Operator of the PDE.
295 \end{methoddesc}
296
297 \begin{methoddesc}[LinearPDE]{getRightHandSide}{}
298 returns the right hand side of the PDE as a \Data object. If
299 \var{ignoreConstraint}=\True, then the constraints are not considered
300 when building up the right hand side.
301 \end{methoddesc}
302
303 \begin{methoddesc}[LinearPDE]{getSystem}{}
304 returns the \Operator and right hand side of the PDE.
305 \end{methoddesc}
306
307 \begin{methoddesc}[LinearPDE]{getSolution}{}
308 returns (an approximation of) the solution of the PDE. This call
309 will invoke the discretization of the PDE and the solution of the resulting
310 system of linear equations. Keep in mind that this call is typically computational
311 expensive and can - depending on the PDE and the discretiztion - take a long time to complete.
312 \end{methoddesc}
313
314
315
316 \subsection{The \Poisson Class}
317 The \Poisson class provides an easy way to define and solve the Poisson
318 equation
319 \begin{equation}\label{POISSON.1}
320 -u\hackscore{,ii}=f\; .
321 \end{equation}
322 with homogeneous boundary conditions
323 \begin{equation}\label{POISSON.2}
324 n\hackscore{i}u\hackscore{,i}=0
325 \end{equation}
326 and homogeneous constraints
327 \begin{equation}\label{POISSON.3}
328 u=0 \mbox{ where } q>0
329 \end{equation}
330 $f$ has to be a \Scalar in the \Function and $q$ must be
331 a \Scalar in the \SolutionFS.
332
333 \begin{classdesc}{Poisson}{domain}
334 opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE.
335 \end{classdesc}
336 \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()}
337 assigns new values to \var{f} and \var{q}.
338 \end{methoddesc}
339
340 \subsection{The \Helmholtz Class}
341 The \Helmholtz class defines the Helmholtz problem
342 \begin{equation}\label{HZ.1}
343 \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f
344 \end{equation}
345 with natural boundary conditions
346 \begin{equation}\label{HZ.2}
347 k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u
348 \end{equation}
349 and constraints:
350 \begin{equation}\label{HZ.3}
351 u=r \mbox{ where } q>0
352 \end{equation}
353 $\omega$, $k$, $f$ have to be a \Scalar in the \Function,
354 $g$ and $\alpha$ must be a \Scalar in the \FunctionOnBoundary,
355 and $q$ and $r$ must be a \Scalar in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace.
356
357 \begin{classdesc}{Helmholtz}{domain}
358 opens a Helmholtz equation on the \Domain domain. \Helmholtz is derived from \LinearPDE.
359 \end{classdesc}
360 \begin{methoddesc}[Helmholtz]{setValue}{ \optional{omega} \optional{, k} \optional{, f} \optional{, alpha} \optional{, g} \optional{, r} \optional{, q}}
361 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.
362 \end{methoddesc}
363
364 \subsection{The \Lame Class}
365 The \Lame class defines a Lame equation problem:
366 \begin{equation}\label{LE.1}
367 -(\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k}\delta\hackscore{ij})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j}
368 \end{equation}
369 with natural boundary conditions:
370 \begin{equation}\label{LE.2}
371 n\hackscore{j}(\mu \; (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k}\delta\hackscore{ij}) = f\hackscore{i}+n\hackscore{j}\sigma\hackscore{ij}
372 \end{equation}
373 and constraint
374 \begin{equation}\label{LE.3}
375 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0
376 \end{equation}
377 $\mu$, $\lambda$ have to be a \Scalar in the \Function,
378 $F$ has to be a \Vector in the \Function,
379 $\sigma$ has to be a \Tensor in the \Function,
380 $f$ must be a \Vector in the \FunctionOnBoundary,
381 and $q$ and $r$ must be a \Vector in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace.
382
383 \begin{classdesc}{Lame}{domain}
384 opens a Lame equation on the \Domain domain. \Lame is derived from \LinearPDE.
385 \end{classdesc}
386 \begin{methoddesc}[Lame]{setValue}{ \optional{lame_lambda} \optional{, lame_mu} \optional{, F} \optional{, sigma} \optional{, f} \optional{, r} \optional{, q}}
387 assigns new values to
388 \var{lame_lambda},
389 \var{lame_mu},
390 \var{F},
391 \var{sigma},
392 \var{f},
393 \var{r} and
394 \var{q}
395 By default all values are set to be zero.
396 \end{methoddesc}
397
398
399
400 \section{Projection}
401 \declaremodule{extension}{esys.escript.pdetools}
402 \label{SEC Projection}
403
404 Using the \LinearPDE class provides an option to change the \FunctionSpace attribute in addition
405 to the standard interpolation mechanism\index{interpolation} as
406 discussed on in Chapter~\ref{ESCRIPT CHAP}. If one looks the
407 stripped down version
408 \begin{equation}\label{PROJ.1}
409 u = Y
410 \end{equation}
411 of the general scalar PDE~\ref{LINEARPDE.SINGLE.1} (boundary conditions are irrelevant)
412 one can see the solution $u$ of this PDE as a project of the input function $Y$
413 which has the \Function attribute to a function with the \SolutionFS or \ReducedSolutionFS
414 attribute. In fact, the solution maps values defined at
415 element centers representing a possibly discontinuous function
416 onto a continuous function represented by its values at the nodes of the FEM mesh.
417 This mapping is called a projection\index{projection}. Projection
418 can be a useful tool but needs to be applied with some care due to the fact that
419 a potentially discontinuous function is projected onto a continuous function but it can
420 also be a desirable effect for instance to smooth a function. The projection of the
421 gradient of a function typically calculated on the element center to the
422 nodes of a FEM mesh can be evaluated on the domain boundary and so projection provides a tool to extrapolate
423 the gradient from the internal to the boundary. This is only a reasonable procedure in the absence of singularities at the boundary.
424
425 As projection is used often in simulations \escript provides an easy to use class \class{Projector}
426 which is part of the \pdetools module. The following script demonstrates
427 the usage of the class to project the piecewise constant function ($=1$ for $x\hackscore{0}\ge 0.5$ and
428 $=-1$ for $x\hackscore{0}<0.5$ ) to a function with the \ReducedSolutionFS attribute (default target)
429 \begin{python}
430 from esys.escript.pdetools import Projector
431 proj=Projector(domain)
432 x0=domain.getX()[0]
433 jmp=1.-2.*wherePositive(x0-0.5)
434 u=proj.getValue(jmp)
435 # alternative call:
436 u=proj(jmp)
437 \end{python}
438 By default the class uses lumping to solve the PDE~\ref{PROJ.1}. This technique is faster
439 then using the standard solver techniques of PDEs. In essence it leads to using the average of
440 neighbor element values to calculate the value at each FEM node.
441
442 The following script illustrate how to evaluate the normal stress
443 on the boundary from a given displacement field \var{u}:
444 \begin{python}
445 from esys.escript.pdetools import Projector
446 u=...
447 proj=Projector(u.getDomin())
448 e=symmetric(grad(u))
449 stress = G*e+ (K-2./3.*G)*trace(e)*kronecker(u.getDomin())
450 normal_stress = inner(u.getDomin().getNormal(), proj(stress))
451 \end{python}
452
453
454
455 \begin{classdesc}{Projector}{domain\optional{, reduce=\True \optional{, fast=\True}}}
456 This class defines the projector on the domain \var{domain}.
457 If \var{reduce} is set to \True the projection will be returned
458 as a \ReducedSolutionFS \Data object. Otherwise \SolutionFS representation is returned.
459 If \var{reduce} is set to \True lumping is used when
460 the equation~\ref{PROJ.1} is solved. Otherwise the standard
461 PDE solver is used. Notice, that lumping is requires significant less
462 compute time and memory. The class is callable.
463 \end{classdesc}
464
465 \begin{methoddesc}[Projector]{getSolverOptions}{}
466 returns the solver options for solving the PDE. In fact the method returns
467 a \SolverOptions class object which can be used to modify the tolerance,
468 the solver or the preconditioner, see Section~\ref{SEC Solver Options} for details.
469 \end{methoddesc}
470
471 \begin{methoddesc}[Projector]{getValue}{input_data}
472 projects the \var{input_data}. This method is equivalent to call an instance
473 of the class with argument \var{input_data}:
474
475 \end{methoddesc}
476
477
478 % \section{Transport Problems}
479 % \label{SEC Transport}
480
481 \section{Solver Options}
482 \label{SEC Solver Options}
483
484 \begin{classdesc}{SolverOptions}{}
485 This class defines the solver options for a linear or non-linear solver.
486 The option also supports the handling of diagnostic informations.
487 \end{classdesc}
488
489 \begin{methoddesc}[SolverOptions]{getSummary}{}
490 Returns a string reporting the current settings
491 \end{methoddesc}
492
493 \begin{methoddesc}[SolverOptions]{getName}{key}
494 Returns the name as a string of a given key
495 \end{methoddesc}
496
497 \begin{methoddesc}[SolverOptions]{setSolverMethod}{\optional{method=SolverOptions.DEFAULT}}
498 Sets the solver method to be used. Use \var{method}=\member{SolverOptions.DIRECT} to indicate that a direct rather than an iterative solver should be used and use \var{method}=\member{SolverOptions.ITERATIVE} to indicate that an iterative rather than a direct solver should be used.
499 The value of \var{method} must be one of the constants
500 \member{SolverOptions.DEFAULT}, \member{SolverOptions.DIRECT}, \member{SolverOptions.CHOLEVSKY}, \member{SolverOptions.PCG},\member{SolverOptions.CR}, \member{SolverOptions.CGS}, \member{SolverOptions.BICGSTAB}, \member{SolverOptions.SSOR},
501 \member{SolverOptions.GMRES}, \member{SolverOptions.PRES20}, \member{SolverOptions.LUMPING}, \member{SolverOptions.ITERATIVE}, \member{SolverOptions.NONLINEAR_GMRES}, \member{SolverOptions.TFQMR}, \member{SolverOptions.MINRES},
502 or \member{SolverOptions.GAUSS_SEIDEL}.
503 Not all packages support all solvers. It can be assumed that a package makes a reasonable choice if it encounters. See Table~\ref{TAB FINLEY SOLVER OPTIONS 1} for the solvers supported by \finley.
504 \end{methoddesc}
505
506 \begin{methoddesc}[SolverOptions]{getSolverMethod}{}
507 Returns key of the solver method to be used.
508 \end{methoddesc}
509
510 \begin{methoddesc}[SolverOptions]{setPreconditioner}{\optional{preconditioner=SolverOptions.JACOBI}}
511 Sets the preconditioner to be used.
512 The value of \var{preconditioner} must be one of the constants
513 \member{SolverOptions.SSOR}, \member{SolverOptions.ILU0}, \member{SolverOptions.ILUT}, \member{SolverOptions.JACOBI},
514 \member{SolverOptions.AMG}, \member{SolverOptions.REC_ILU}, \member{SolverOptions.GAUSS_SEIDEL}, \member{SolverOptions.RILU}, or
515 \member{SolverOptions.NO_PRECONDITIONER}.
516 Not all packages support all preconditioner. It can be assumed that a package makes a reasonable choice if it encounters
517 an unknown preconditioner. See Table~\ref{TAB FINLEY SOLVER OPTIONS 2} for the solvers supported by \finley.
518 \end{methoddesc}
519
520 \begin{methoddesc}[SolverOptions]{getPreconditioner}{}
521 Returns key of the preconditioner to be used.
522 \end{methoddesc}
523
524 \begin{methoddesc}[SolverOptions]{setPackage}{\optional{package=SolverOptions.DEFAULT}}
525 Sets the solver package to be used as a solver.
526 The value of \var{method} must be one of the constants in \member{SolverOptions.DEFAULT}, \member{SolverOptions.PASO}, \member{SolverOptions.SUPER_LU}, \member{SolverOptions.PASTIX}, \member{SolverOptions.MKL}, \member{SolverOptions.UMFPACK}, \member{SolverOptions.TRILINOS}.
527 Not all packages are support on all implementation. An exception may be thrown on some platforms if a particular package is requested. Currently \finley supports \member{SolverOptions.PASO} (as default)
528 and, if available, \member{SolverOptions.MKL}
529 \footnote{If the stiffness matrix is non-regular \MKL may return without
530 returning a proper error code. If you observe suspicious solutions when using MKL, this may cause by a non-invertible operator. }
531 and \member{SolverOptions.UMFPACK}
532
533 \end{methoddesc}
534
535 \begin{methoddesc}[SolverOptions]{getPackage}{}
536 Returns the solver package key
537 \end{methoddesc}
538
539
540 \begin{methoddesc}[SolverOptions]{resetDiagnostics}{\optional{all=False}}
541 resets the diagnostics. If \var{all} is \True all diagnostics including accumulative counters are reset.
542 \end{methoddesc}
543
544 \begin{methoddesc}[SolverOptions]{getDiagnostics}{\optional{ name}}
545 Returns the diagnostic information \var{name}. The following keywords are
546 supported:
547 \begin{itemize}
548 \item "num_iter": the number of iteration steps
549 \item "cum_num_iter": the cumulative number of iteration steps
550 \item "num_level": the number of level in multi level solver
551 \item "num_inner_iter": the number of inner iteration steps
552 \item"cum_num_inner_iter": the cumulative number of inner iteration steps
553 \item"time": execution time
554 \item "cum_time": cumulative execution time
555 \item "set_up_time": time to set up of the solver, typically this includes factorization and reordering
556 \item "cum_set_up_time": cumulative time to set up of the solver
557 \item "net_time": net execution time, excluding setup time for the solver and execution time for preconditioner
558 \item "cum_net_time": cumulative net execution time
559 \item "residual_norm": norm of the final residual
560 \item "converged": return self.__converged
561 \end{itemize}
562 \end{methoddesc}
563
564
565 \begin{methoddesc}[SolverOptions]{hasConverged}{}
566 Returns \True if the last solver call has been finalized successfully.
567 If an exception has been thrown by the solver the status of this flag is undefined.
568 \end{methoddesc}
569
570 \begin{methoddesc}[SolverOptions]{setCoarsening}{\optional{method=SolverOptions.DEFAULT}}
571 Sets the key of the coarsening method to be applied in \AMG.
572 The value of \var{method} must be one of the constants
573 \member{SolverOptions.DEFAULT}
574 \member{SolverOptions.STANDARD_COARSENING}
575 \member{SolverOptions.YAIR_SHAPIRA_COARSENING}, \\
576 \member{SolverOptions.RUGE_STUEBEN_COARSENING}~\footnote{The Ruge-Stueben and aggregation coarsening algorithms used for measuring the strength of connection only, but splitting is done with greedy algorithm.}, \\or \member{SolverOptions.AGGREGATION_COARSENING}.
577 \end{methoddesc}
578
579 \begin{methoddesc}[SolverOptions]{getCoarsening}{}
580 Returns the key of the coarsening algorithm to be applied \AMG.
581 \end{methoddesc}
582
583 \begin{methoddesc}[SolverOptions]{setReordering}{\optional{ordering=SolverOptions.DEFAULT_REORDERING}}
584 Sets the key of the reordering method to be applied if supported by the solver. Some direct solvers support reordering to optimize compute time and storage use during elimination. The value of \var{ordering} must be one of the constants
585 \member{SolverOptions.NO_REORDERING}, \member{SolverOptions.MINIMUM_FILL_IN},
586 \member{SolverOptions.NESTED_DISSECTION}, or \member{SolverOptions.DEFAULT_REORDERING}.
587 \end{methoddesc}
588
589 \begin{methoddesc}[SolverOptions]{getReordering}{}
590 Returns the key of the reordering method to be applied if supported by the solver.
591 \end{methoddesc}
592
593 \begin{methoddesc}[SolverOptions]{setRestart}{\optional{restart=None}}
594 Sets the number of iterations steps after which \GMRES is performing a restart.
595 If \var{restart} is equal to \var{None} no restart is performed.
596 \end{methoddesc}
597
598
599 \begin{methoddesc}[SolverOptions]{getRestart}{}
600 Returns the number of iterations steps after which \GMRES is performing a restart.
601 \end{methoddesc}
602
603 \begin{methoddesc}[SolverOptions]{setTruncation}{\optional{truncation=20}}
604 Sets the number of residuals in \GMRES to be stored for orthogonalization. The more residuals are stored the faster \GMRES converged but
605 \end{methoddesc}
606
607 \begin{methoddesc}[SolverOptions]{getTruncation}{}
608 Returns the number of residuals in \GMRES to be stored for orthogonalization
609 \end{methoddesc}
610
611
612 \begin{methoddesc}[SolverOptions]{setIterMax}{\optional{iter_max=10000}}
613 Sets the maximum number of iteration steps
614 \end{methoddesc}
615
616 \begin{methoddesc}[SolverOptions]{getIterMax}{}
617 Returns maximum number of iteration steps
618 \end{methoddesc}
619
620 \begin{methoddesc}[SolverOptions]{setLevelMax}{\optional{level_max=10}}
621 Sets the maximum number of coarsening levels to be used in the \AMG solver or preconditioner.
622 \end{methoddesc}
623
624 \begin{methoddesc}[SolverOptions]{getLevelMax}{}
625 Returns the maximum number of coarsening levels to be used in an algebraic multi level solver or preconditioner
626 \end{methoddesc}
627
628 \begin{methoddesc}[SolverOptions]{setCoarseningThreshold}{\optional{theta=0.25}}
629 Sets the threshold for coarsening in the \AMG solver or preconditioner
630 \end{methoddesc}
631
632 \begin{methoddesc}[SolverOptions]{getCoarseningThreshold}{}
633 Returns the threshold for coarsening in the \AMG solver or preconditioner
634 \end{methoddesc}
635
636 \begin{methoddesc}[SolverOptions]{setMinCoarseMatrixSize}{\optional{size=500}}
637 Sets the minumum size of the coarsest level matrix in \AMG.
638 \end{methoddesc}
639
640 \begin{methoddesc}[SolverOptions]{getMinCoarseMatrixSize}{}
641 Returns the minumum size of the coarsest level matrix in \AMG.
642 \end{methoddesc}
643
644 \begin{methoddesc}[SolverOptions]{setSmoother}{\optional{smoother=\GAUSSSEIDEL}}
645 Sets the \JACOBI or \GAUSSSEIDEL smoother to be used in \AMG.
646 \end{methoddesc}
647
648 \begin{methoddesc}[SolverOptions]{getSmoother}{}
649 Returns the key for \JACOBI or \GAUSSSEIDEL smoother used in \AMG.
650 \end{methoddesc}
651
652 \begin{methoddesc}[SolverOptions]{setNumSweeps}{\optional{sweeps=2}}
653 Sets the number of sweeps in a \JACOBI or \GAUSSSEIDEL preconditioner.
654 \end{methoddesc}
655
656 \begin{methoddesc}[SolverOptions]{getNumSweeps}{}
657 Returns the number of sweeps in a \JACOBI or \GAUSSSEIDEL preconditioner.
658 \end{methoddesc}
659
660 \begin{methoddesc}[SolverOptions]{setNumPreSweeps}{\optional{sweeps=2}}
661 Sets the number of sweeps in the pre-smoothing step of \AMG
662 \end{methoddesc}
663
664 \begin{methoddesc}[SolverOptions]{getNumPreSweeps}{}
665 Returns the number of sweeps in the pre-smoothing step of \AMG
666 \end{methoddesc}
667
668 \begin{methoddesc}[SolverOptions]{setNumPostSweeps}{\optional{sweeps=2}}
669 Sets the number of sweeps in the post-smoothing step of \AMG
670 \end{methoddesc}
671
672 \begin{methoddesc}[SolverOptions]{getNumPostSweeps}{}
673 Returns he number of sweeps sweeps in the post-smoothing step of \AMG
674 \end{methoddesc}
675
676 \begin{methoddesc}[SolverOptions]{setTolerance}{\optional{rtol=1.e-8}}
677 Sets the relative tolerance for the solver. The actually meaning of tolerance depends
678 on the underlying PDE library. In most cases, the tolerance
679 will only consider the error from solving the discrete problem but will
680 not consider any discretization error.
681 \end{methoddesc}
682
683 \begin{methoddesc}[SolverOptions]{getTolerance}{}
684 Returns the relative tolerance for the solver
685 \end{methoddesc}
686
687 \begin{methoddesc}[SolverOptions]{setAbsoluteTolerance}{\optional{atol=0.}}
688 Sets the absolute tolerance for the solver. The actually meaning of tolerance depends
689 on the underlying PDE library. In most cases, the tolerance
690 will only consider the error from solving the discrete problem but will
691 not consider any discretization error.
692 \end{methoddesc}
693
694 \begin{methoddesc}[SolverOptions]{getAbsoluteTolerance}{}
695 Returns the absolute tolerance for the solver
696 \end{methoddesc}
697
698
699 \begin{methoddesc}[SolverOptions]{setInnerTolerance}{\optional{rtol=0.9}}
700 Sets the relative tolerance for an inner iteration scheme for instance
701 on the coarsest level in a multi-level scheme.
702 \end{methoddesc}
703
704 \begin{methoddesc}[SolverOptions]{getInnerTolerance}{}
705 Returns the relative tolerance for an inner iteration scheme
706 \end{methoddesc}
707
708 \begin{methoddesc}[SolverOptions]{setDropTolerance}{\optional{drop_tol=0.01}}
709 Sets the relative drop tolerance in ILUT
710 \end{methoddesc}
711
712 \begin{methoddesc}[SolverOptions]{getDropTolerance}{}
713 Returns the relative drop tolerance in \ILUT
714 \end{methoddesc}
715
716
717 \begin{methoddesc}[SolverOptions]{setDropStorage}{\optional{storage=2.}}
718 Sets the maximum allowed increase in storage for \ILUT. \var{storage}=2 would mean that a doubling of the storage needed for the coefficient matrix is allowed in the \ILUT factorization.
719 \end{methoddesc}
720
721 \begin{methoddesc}[SolverOptions]{getDropStorage}{}
722 Returns the maximum allowed increase in storage for \ILUT
723 \end{methoddesc}
724
725 \begin{methoddesc}[SolverOptions]{setRelaxationFactor}{\optional{factor=0.3}}
726 Sets the relaxation factor used to add dropped elements in \RILU to the main diagonal.
727 \end{methoddesc}
728
729 \begin{methoddesc}[SolverOptions]{getRelaxationFactor}{}
730 Returns the relaxation factor used to add dropped elements in RILU to the main diagonal.
731 \end{methoddesc}
732
733 \begin{methoddesc}[SolverOptions]{isSymmetric}{}
734 Returns \True is the descrete system is indicated as symmetric.
735 \end{methoddesc}
736
737 \begin{methoddesc}[SolverOptions]{setSymmetryOn}{}
738 Sets the symmetry flag to indicate that the coefficient matrix is symmetric.
739 \end{methoddesc}
740
741 \begin{methoddesc}[SolverOptions]{setSymmetryOff}{}
742 Clears the symmetry flag for the coefficient matrix.
743 \end{methoddesc}
744
745 \begin{methoddesc}[SolverOptions]{isVerbose}{}
746 Returns \True if the solver is expected to be verbose.
747 \end{methoddesc}
748
749
750 \begin{methoddesc}[SolverOptions]{setVerbosityOn}{}
751 Switches the verbosity of the solver on.
752 \end{methoddesc}
753
754
755 \begin{methoddesc}[SolverOptions]{setVerbosityOff}{}
756 Switches the verbosity of the solver off.
757 \end{methoddesc}
758
759
760 \begin{methoddesc}[SolverOptions]{adaptInnerTolerance}{}
761 Returns \True if the tolerance of the inner solver is selected automatically.
762 Otherwise the inner tolerance set by \member{setInnerTolerance} is used.
763 \end{methoddesc}
764
765 \begin{methoddesc}[SolverOptions]{setInnerToleranceAdaptionOn}{}
766 Switches the automatic selection of inner tolerance on
767 \end{methoddesc}
768
769 \begin{methoddesc}[SolverOptions]{setInnerToleranceAdaptionOff}{}
770 Switches the automatic selection of inner tolerance off.
771 \end{methoddesc}
772
773 \begin{methoddesc}[SolverOptions]{setInnerIterMax}{\optional{iter_max=10}}
774 Sets the maximum number of iteration steps for the inner iteration.
775 \end{methoddesc}
776
777 \begin{methoddesc}[SolverOptions]{getInnerIterMax}{}
778 Returns maximum number of inner iteration steps.
779 \end{methoddesc}
780
781 \begin{methoddesc}[SolverOptions]{acceptConvergenceFailure}{}
782 Returns \True if a failure to meet the stopping criteria within the
783 given number of iteration steps is not raising in exception. This is useful
784 if a solver is used in a non-linear context where the non-linear solver can
785 continue even if the returned the solution does not necessarily meet the
786 stopping criteria. One can use the \member{hasConverged} method to check if the
787 last call to the solver was successful.
788 \end{methoddesc}
789
790 \begin{methoddesc}[SolverOptions]{setAcceptanceConvergenceFailureOn}{}
791 Switches the acceptance of a failure of convergence on.
792 \end{methoddesc}
793
794 \begin{methoddesc}[SolverOptions]{setAcceptanceConvergenceFailureOff}{}
795 Switches the acceptance of a failure of convergence off.
796 \end{methoddesc}
797
798 \begin{memberdesc}[SolverOptions]{DEFAULT}
799 default method, preconditioner or package to be used to solve the PDE. An appropriate method should be
800 chosen by the used PDE solver library.
801 \end{memberdesc}
802
803 \begin{memberdesc}[SolverOptions]{MKL}
804 the \MKL library by Intel,~\Ref{MKL}\footnote{The \MKL library will only be available when the Intel compilation environment is used.}.
805 \end{memberdesc}
806
807 \begin{memberdesc}[SolverOptions]{UMFPACK}
808 the \UMFPACK,~\Ref{UMFPACK}. Remark: \UMFPACK is not parallelized.
809 \end{memberdesc}
810
811 \begin{memberdesc}[SolverOptions]{PASO}
812 \PASO is the solver library of \finley, see \Sec{CHAPTER ON FINLEY}.
813 \end{memberdesc}
814
815 \begin{memberdesc}[SolverOptions]{ITERATIVE}
816 the default iterative method and preconditioner. The actually used method depends on the PDE solver library and the solver package been chosen. Typically, \PCG is used for symmetric PDEsand \BiCGStab otherwise, both with \JACOBI preconditioner.
817 \end{memberdesc}
818
819 \begin{memberdesc}[SolverOptions]{DIRECT}
820 the default direct linear solver.
821 \end{memberdesc}
822
823 \begin{memberdesc}[SolverOptions]{CHOLEVSKY}
824 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE.
825 \end{memberdesc}
826
827 \begin{memberdesc}[SolverOptions]{PCG}
828 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE.
829 \end{memberdesc}
830
831 \begin{memberdesc}[SolverOptions]{TFQMR}
832 transpose-free quasi-minimal residual method, see~\Ref{WEISS}\index{linear solver!TFQMR}\index{TFQMR}. \end{memberdesc}
833
834 \begin{memberdesc}[SolverOptions]{GMRES}
835 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters
836 \var{truncation} and \var{restart} of \method{getSolution}.
837 \end{memberdesc}
838
839 \begin{memberdesc}[SolverOptions]{MINRES}
840 minimal residual method method, \index{linear solver!MINRES}\index{MINRES} \end{memberdesc}
841
842 \begin{memberdesc}[SolverOptions]{LUMPING}
843 uses lumping to solve the system of linear equations~\index{linear solver!lumping}\index{lumping}. This solver technique
844 condenses the stiffness matrix to a diagonal matrix so the solution of the linear systems becomes very cheap. It can be used when
845 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
846 but is expected to converge to zero when the mesh gets finer.
847 Lumping does not use the linear system solver library.
848 \end{memberdesc}
849
850 \begin{memberdesc}[SolverOptions]{PRES20}
851 the GMRES method with truncation after five residuals and
852 restart after 20 steps, see~\Ref{WEISS}.
853 \end{memberdesc}
854
855 \begin{memberdesc}[SolverOptions]{CGS}
856 conjugate gradient squared method, see~\Ref{WEISS}.
857 \end{memberdesc}
858
859 \begin{memberdesc}[SolverOptions]{BICGSTAB}
860 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}.
861 \end{memberdesc}
862
863 \begin{memberdesc}[SolverOptions]{SSOR}
864 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support
865 this as a solver.
866 \end{memberdesc}
867
868 \begin{memberdesc}[SolverOptions]{ILU0}
869 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}.
870 \end{memberdesc}
871
872 \begin{memberdesc}[SolverOptions]{ILUT}
873 the incomplete LU factorization preconditioner with fill-in, see~\Ref{Saad}. During the LU-factorization element with
874 relative size less then \member{getDropTolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the
875 \member{getDropStorage}-fold of the stiffness matrix. \member{getDropTolerance} and \member{getDropStorage} are both set in the
876 \method{getSolution} call.
877 \end{memberdesc}
878
879 \begin{memberdesc}[SolverOptions]{JACOBI}
880 the Jacobi preconditioner, see~\Ref{Saad}.
881 \end{memberdesc}
882
883
884 \begin{memberdesc}[SolverOptions]{AMG}
885 the algebraic--multi grid method, see~\Ref{AMG}. This method can be used as linear solver method but is more robust when used
886 in a preconditioner.
887 \end{memberdesc}
888
889 \begin{memberdesc}[SolverOptions]{GAUSS_SEIDEL}
890 the symmetric Gauss-Seidel preconditioner, see~\Ref{Saad}.
891 \member{getNumSweeps()} is the number of sweeps used.
892 \end{memberdesc}
893
894 \begin{memberdesc}[SolverOptions]{RILU}
895 relaxed incomplete LU factorization preconditioner, see~\Ref{RELAXILU}. This method is similar to \ILU0 but dropped elements are added to the main diagonal
896 with the relaxation factor \member{getRelaxationFactor}
897 \end{memberdesc}
898
899 \begin{memberdesc}[SolverOptions]{REC_ILU}
900 recursive incomplete LU factorization preconditioner, see~\Ref{RILU}. This method is similar to \ILU0 but applies reordering during the factorization.
901 \end{memberdesc}
902
903 \begin{memberdesc}[SolverOptions]{NO_REORDERING}
904 no ordering is used during factorization.
905 \end{memberdesc}
906
907 \begin{memberdesc}[SolverOptions]{DEFAULT_REORDERING}
908 the default reordering method during factorization.
909 \end{memberdesc}
910
911 \begin{memberdesc}[SolverOptions]{MINIMUM_FILL_IN}
912 applies reordering before factorization using a fill-in minimization strategy. You have to check with the particular solver library or
913 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in.
914 \end{memberdesc}
915
916 \begin{memberdesc}[SolverOptions]{NESTED_DISSECTION}
917 applies reordering before factorization using a nested dissection strategy. You have to check with the particular solver library or
918 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in.
919 \end{memberdesc}
920
921 \begin{memberdesc}[SolverOptions]{TRILINOS}
922 the Trilinos library is used as a solver~\Ref{TRILINOS}
923 \end{memberdesc}
924
925 \begin{memberdesc}[SolverOptions]{SUPER_LU}
926 the SuperLU library is used as a solver~\Ref{SuperLU}
927 \end{memberdesc}
928
929 \begin{memberdesc}[SolverOptions]{PASTIX}
930 the Pastix library is used as a solver~\Ref{PASTIX}
931 \end{memberdesc}
932
933
934 \begin{memberdesc}[SolverOptions]{STANDARD_COARSENING}
935 \AMG coarsening method by Ruge and Stueben using measure of importance principle~\cite{Multigrid}.
936 \end{memberdesc}
937
938 \begin{memberdesc}[SolverOptions]{YAIR_SHAPIRA_COARSENING}
939 \AMG coarsening method by Yair-Shapira
940 \end{memberdesc}
941
942 \begin{memberdesc}[SolverOptions]{RUGE_STUEBEN_COARSENING} \AMG coarsening method by Ruge and Stueben using greedy algorithm for splitting.
943 \end{memberdesc}
944
945 \begin{memberdesc}[SolverOptions]{AGGREGATION_COARSENING} \AMG coarsening using (symmetric) aggregation using greedy algorithm for splitting.
946 \end{memberdesc}
947
948 \begin{memberdesc}[SolverOptions]{NO_PRECONDITIONER}
949 no preconditioner is applied.
950 \end{memberdesc}
951

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