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 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2010 by University of Queensland 5 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 % 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 14 15 \chapter{The \linearPDEs Module} 16 17 \section{Linear Partial Differential Equations} 18 \label{SEC LinearPDE} 19 20 The \LinearPDE class is used to define a general linear, steady, second order PDE 21 for an unknown function $u$ on a given $\Omega$ defined through a \Domain object. 22 In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes 23 the outer normal field on $\Gamma$. 24 25 For a single PDE with a solution with a single component the linear PDE is defined in the 26 following form: 27 \begin{equation}\label{LINEARPDE.SINGLE.1} 28 -(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 \; . 29 \end{equation} 30 $u_{,j}$ 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. 31 The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the 32 \Function on the PDE or objects that can be converted into such \Data objects. 33 $A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar. 34 The following natural 35 boundary conditions are considered \index{boundary condition!natural} on $\Gamma$: 36 \begin{equation}\label{LINEARPDE.SINGLE.2} 37 n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;. 38 \end{equation} 39 Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are 40 each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the 41 solution at certain locations in the domain. They have the form 42 \begin{equation}\label{LINEARPDE.SINGLE.3} 43 u=r \mbox{ where } q>0 44 \end{equation} 45 $r$ and $q$ are each \Scalar where $q$ is the characteristic function 46 \index{characteristic function} defining where the constraint is applied. 47 The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1} 48 or \eqn{LINEARPDE.SINGLE.2}. 49 50 For a system of PDEs and a solution with several components the PDE has the form 51 \begin{equation}\label{LINEARPDE.SYSTEM.1} 52 -(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} \; . 53 \end{equation} 54 $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne. 55 The natural boundary conditions \index{boundary condition!natural} take the form: 56 \begin{equation}\label{LINEARPDE.SYSTEM.2} 57 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} \;. 58 \end{equation} 59 The coefficient $d$ is a \RankTwo and $y$ is a 60 \RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form 61 \begin{equation}\label{LINEARPDE.SYSTEM.3} 62 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 63 \end{equation} 64 $r$ and $q$ are each \RankOne. Notice that not necessarily all components must 65 have a constraint at all locations. 66 67 \LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$ 68 in the domain $\Omega$. To specify the conditions across the discontinuity we are using the 69 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 70 defined as 71 \begin{equation}\label{LINEARPDE.SYSTEM.5} 72 J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij} 73 \end{equation} 74 For the case of single solution component and single PDE $J$ is defined 75 \begin{equation}\label{LINEARPDE.SINGLE.5} 76 J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j} 77 \end{equation} 78 In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the 79 discontinuity pointing from side 0 towards side 1. For a system of PDEs 80 the contact condition takes the form 81 \begin{equation}\label{LINEARPDE.SYSTEM.6} 82 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} \; . 83 \end{equation} 84 where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the 85 discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference 86 of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. 87 The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a 88 \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne. 89 In case of a single PDE and a single component solution the contact condition takes the form 90 \begin{equation}\label{LINEARPDE.SINGLE.6} 91 n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u] 92 \end{equation} 93 In this case the the coefficient $d^{contact}$ and $y^{contact}$ are each \Scalar 94 both in the \FunctionOnContactZero or \FunctionOnContactOne. 95 96 The PDE is symmetrical \index{symmetrical} if 97 \begin{equation}\label{LINEARPDE.SINGLE.4} 98 A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j} 99 \end{equation} 100 The system of PDEs is symmetrical \index{symmetrical} if 101 \begin{eqnarray} 102 \label{LINEARPDE.SYSTEM.4} 103 A\hackscore{ijkl}&=&A\hackscore{klij} \\ 104 B\hackscore{ijk}&=&C\hackscore{kij} \\ 105 D\hackscore{ik}&=&D\hackscore{ki} \\ 106 d\hackscore{ik}&=&d\hackscore{ki} \\ 107 d^{contact}\hackscore{ik}&=&d^{contact}\hackscore{ki} 108 \end{eqnarray} 109 Note that in contrast with the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$ 110 have to be inspected. 111 112 The following example illustrates the typical usage of the \LinearPDE class: 113 \begin{python} 114 from esys.escript import * 115 from esys.escript.linearPDEs import LinearPDE 116 from esys.finley import Rectangle 117 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 118 mypde=LinearPDE(mydomain) 119 mypde.setSymmetryOn() 120 mypde.setValue(A=kappa*kronecker(mydomain),D=1,Y=1) 121 u=mypde.getSolution() 122 \end{python} 123 We refer to chapter~\ref{CHAP: Tutorial} for more details. 124 125 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 126 code the \method{getSolverOptions} is used to access the \SolverOptions 127 attached to \var{mypde}: 128 \begin{python} 129 from esys.escript import * 130 from esys.escript.linearPDEs import LinearPDE, SolverOptions 131 from esys.finley import Rectangle 132 mydomain = Rectangle(l0=1.,l1=1.,n0=40, n1=20) 133 mypde=LinearPDE(mydomain) 134 mypde.setValue(A=kappa*kronecker(mydomain),D=1,Y=1) 135 mypde.getSolverOptions().setVerbosityOn() 136 mypde.getSolverOptions().setSolverMethod(SolverOptions.PCG) 137 mypde.getSolverOptions().setPreconditioner(SolverOptions.AMG) 138 mypde.getSolverOptions().setTolerance(1e-8) 139 mypde.getSolverOptions().setIterMax(1000) 140 u=mypde.getSolution() 141 \end{python} 142 In this code the preconditioned conjugate gradient method \PCG 143 with preconditioner \AMG. The relative tolerance is set to $10^{-8}$ and 144 the maximum number of iteration steps to $1000$. 145 146 Moreover, after a completed solution call 147 the attached \SolverOptions object gives access to diagnostic informations: 148 \begin{python} 149 u=mypde.getSolution() 150 print 'Number of iteration steps =', mypde.getDiagnostics('num_iter') 151 print 'Total solution time =', mypde.getDiagnostics('time') 152 print 'Set-up time =', mypde.getDiagnostics('set_up_time') 153 print 'Net time =', mypde.getDiagnostics('net_time') 154 print 'Residual norm of returned solution =', mypde.getDiagnostics('residual_norm') 155 \end{python} 156 Typically a negative value for a diagnostic value indicates that the value is undefined. 157 158 \subsection{Classes} 159 \declaremodule{extension}{esys.escript.linearPDEs} 160 \modulesynopsis{Linear partial differential equation handler} 161 The module \linearPDEs provides an interface to define and solve linear partial 162 differential equations within \escript. The module \linearPDEs does not provide any 163 solver capabilities in itself but hands the PDE over to 164 the PDE solver library defined through the \Domain of the PDE, eg. \finley. 165 The general interface is provided through the \LinearPDE class. The \Poisson 166 class which is also derived form the \LinearPDE class should be used 167 to define the Poisson equation \index{Poisson}. 168 169 \subsection{\LinearPDE class} 170 This is the general class to define a linear PDE in \escript. We list a selection of the most 171 important methods of the class. For a complete list, see the reference at \ReferenceGuide. 172 173 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0} 174 opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations} 175 and \var{numSolutions} gives the number of equations and the number of solution components. 176 If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations 177 and the number solutions, respectively, stay undefined until a coefficient is 178 defined. 179 \end{classdesc} 180 181 \subsubsection{\LinearPDE methods} 182 183 \begin{methoddesc}[LinearPDE]{setValue}{ 184 \optional{A}\optional{, B}, 185 \optional{, C}\optional{, D} 186 \optional{, X}\optional{, Y} 187 \optional{, d}\optional{, y} 188 \optional{, d_contact}\optional{, y_contact} 189 \optional{, q}\optional{, r}} 190 assigns new values to coefficients. By default all values are assumed to be zero\footnote{ 191 In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The 192 can by used by the solver library to reduce computational costs. 193 } 194 If the new coefficient value is not a \Data object, it is converted into a \Data object in the 195 appropriate \FunctionSpace. 196 \end{methoddesc} 197 198 \begin{methoddesc}[LinearPDE]{getCoefficient}{name} 199 return the value assigned to coefficient \var{name}. If \var{name} is not a valid name 200 an exception is raised. 201 \end{methoddesc} 202 203 \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name} 204 returns the shape of coefficient \var{name} even if no value has been assigned to it. 205 \end{methoddesc} 206 207 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name} 208 returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it. 209 \end{methoddesc} 210 211 \begin{methoddesc}[LinearPDE]{setDebugOn}{} 212 switches on debug mode. 213 \end{methoddesc} 214 215 \begin{methoddesc}[LinearPDE]{setDebugOff}{} 216 switches off debug mode. 217 \end{methoddesc} 218 219 \begin{methoddesc}[LinearPDE]{getSolverOptions}{} 220 returns the solver options for solving the PDE. In fact the method returns 221 a \SolverOptions class object which can be used to modify the tolerance, 222 the solver or the preconditioner, see Section~\ref{SEC Solver Options} for details. 223 \end{methoddesc} 224 225 \begin{methoddesc}[LinearPDE]{setSolverOptions}{\optional{options=None}} 226 sets the solver options for solving the PDE. If argument \var{options} is present it 227 must be a \SolverOptions class object, see Section~\ref{SEC Solver Options} for details. Otherwise the solver options are reset to the default. 228 \end{methoddesc} 229 230 231 \begin{methoddesc}[LinearPDE]{isUsingLumping}{} 232 returns \True if \LUMPING is set as the solver for the system of linear equations. 233 Otherwise \False is returned. 234 \end{methoddesc} 235 236 237 \begin{methoddesc}[LinearPDE]{getDomain}{} 238 returns the \Domain of the PDE. 239 \end{methoddesc} 240 241 \begin{methoddesc}[LinearPDE]{getDim}{} 242 returns the spatial dimension of the PDE. 243 \end{methoddesc} 244 245 \begin{methoddesc}[LinearPDE]{getNumEquations}{} 246 returns the number of equations. 247 \end{methoddesc} 248 249 \begin{methoddesc}[LinearPDE]{getNumSolutions}{} 250 returns the number of components of the solution. 251 \end{methoddesc} 252 253 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False} 254 returns \True if the PDE is symmetric and \False otherwise. 255 The method is very computationally expensive and should only be 256 called for testing purposes. The symmetry flag is not altered. 257 If \var{verbose}=\True information about where symmetry is violated 258 are printed. 259 \end{methoddesc} 260 261 \begin{methoddesc}[LinearPDE]{getFlux}{u} 262 returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u} 263 defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively. 264 \end{methoddesc} 265 266 267 \begin{methoddesc}[LinearPDE]{isSymmetric}{} 268 returns \True if the PDE has been indicated to be symmetric. 269 Otherwise \False is returned. 270 \end{methoddesc} 271 272 \begin{methoddesc}[LinearPDE]{setSymmetryOn}{} 273 indicates that the PDE is symmetric. 274 \end{methoddesc} 275 276 \begin{methoddesc}[LinearPDE]{setSymmetryOff}{} 277 indicates that the PDE is not symmetric. 278 \end{methoddesc} 279 280 \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{} 281 switches on the reduction of polynomial order for the solution and equation evaluation even if 282 a quadratic or higher interpolation order is defined in the \Domain. This feature may not 283 be supported by all PDE libraries. 284 \end{methoddesc} 285 286 \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{} 287 switches off the reduction of polynomial order for the solution and 288 equation evaluation. 289 \end{methoddesc} 290 291 \begin{methoddesc}[LinearPDE]{getOperator}{} 292 returns the \Operator of the PDE. 293 \end{methoddesc} 294 295 \begin{methoddesc}[LinearPDE]{getRightHandSide}{} 296 returns the right hand side of the PDE as a \Data object. If 297 \var{ignoreConstraint}=\True, then the constraints are not considered 298 when building up the right hand side. 299 \end{methoddesc} 300 301 \begin{methoddesc}[LinearPDE]{getSystem}{} 302 returns the \Operator and right hand side of the PDE. 303 \end{methoddesc} 304 305 \begin{methoddesc}[LinearPDE]{getSolution}{} 306 returns (an approximation of) the solution of the PDE. This call 307 will invoke the discretization of the PDE and the solution of the resulting 308 system of linear equations. Keep in mind that this call is typically computational 309 expensive and can - depending on the PDE and the discretiztion - take a long time to complete. 310 \end{methoddesc} 311 312 313 314 \subsection{The \Poisson Class} 315 The \Poisson class provides an easy way to define and solve the Poisson 316 equation 317 \begin{equation}\label{POISSON.1} 318 -u\hackscore{,ii}=f\; . 319 \end{equation} 320 with homogeneous boundary conditions 321 \begin{equation}\label{POISSON.2} 322 n\hackscore{i}u\hackscore{,i}=0 323 \end{equation} 324 and homogeneous constraints 325 \begin{equation}\label{POISSON.3} 326 u=0 \mbox{ where } q>0 327 \end{equation} 328 $f$ has to be a \Scalar in the \Function and $q$ must be 329 a \Scalar in the \SolutionFS. 330 331 \begin{classdesc}{Poisson}{domain} 332 opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE. 333 \end{classdesc} 334 \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()} 335 assigns new values to \var{f} and \var{q}. 336 \end{methoddesc} 337 338 \subsection{The \Helmholtz Class} 339 The \Helmholtz class defines the Helmholtz problem 340 \begin{equation}\label{HZ.1} 341 \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f 342 \end{equation} 343 with natural boundary conditions 344 \begin{equation}\label{HZ.2} 345 k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u 346 \end{equation} 347 and constraints: 348 \begin{equation}\label{HZ.3} 349 u=r \mbox{ where } q>0 350 \end{equation} 351 $\omega$, $k$, $f$ have to be a \Scalar in the \Function, 352 $g$ and $\alpha$ must be a \Scalar in the \FunctionOnBoundary, 353 and $q$ and $r$ must be a \Scalar in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 354 355 \begin{classdesc}{Helmholtz}{domain} 356 opens a Helmholtz equation on the \Domain domain. \Helmholtz is derived from \LinearPDE. 357 \end{classdesc} 358 \begin{methoddesc}[Helmholtz]{setValue}{ \optional{omega} \optional{, k} \optional{, f} \optional{, alpha} \optional{, g} \optional{, r} \optional{, q}} 359 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. 360 \end{methoddesc} 361 362 \subsection{The \Lame Class} 363 The \Lame class defines a Lame equation problem: 364 \begin{equation}\label{LE.1} 365 -(\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} 366 \end{equation} 367 with natural boundary conditions: 368 \begin{equation}\label{LE.2} 369 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} 370 \end{equation} 371 and constraint 372 \begin{equation}\label{LE.3} 373 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 374 \end{equation} 375 $\mu$, $\lambda$ have to be a \Scalar in the \Function, 376 $F$ has to be a \Vector in the \Function, 377 $\sigma$ has to be a \Tensor in the \Function, 378 $f$ must be a \Vector in the \FunctionOnBoundary, 379 and $q$ and $r$ must be a \Vector in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 380 381 \begin{classdesc}{Lame}{domain} 382 opens a Lame equation on the \Domain domain. \Lame is derived from \LinearPDE. 383 \end{classdesc} 384 \begin{methoddesc}[Lame]{setValue}{ \optional{lame_lambda} \optional{, lame_mu} \optional{, F} \optional{, sigma} \optional{, f} \optional{, r} \optional{, q}} 385 assigns new values to 386 \var{lame_lambda}, 387 \var{lame_mu}, 388 \var{F}, 389 \var{sigma}, 390 \var{f}, 391 \var{r} and 392 \var{q} 393 By default all values are set to be zero. 394 \end{methoddesc} 395 396 397 398 \section{Projection} 399 \declaremodule{extension}{esys.escript.pdetools} 400 \label{SEC Projection} 401 402 Using the \LinearPDE class provides an option to change the \FunctionSpace attribute in addition 403 to the standard interpolation mechanism\index{interpolation} as 404 discussed on in Chapter~\ref{ESCRIPT CHAP}. If one looks the 405 stripped down version 406 \begin{equation}\label{PROJ.1} 407 u = Y 408 \end{equation} 409 of the general scalar PDE~\ref{LINEARPDE.SINGLE.1} (boundary conditions are irrelevant) 410 one can see the solution $u$ of this PDE as a project of the input function $Y$ 411 which has the \Function attribute to a function with the \SolutionFS or \ReducedSolutionFS 412 attribute. In fact, the solution maps values defined at 413 element centers representing a possibly discontinuous function 414 onto a continuous function represented by its values at the nodes of the FEM mesh. 415 This mapping is called a projection\index{projection}. Projection 416 can be a useful tool but needs to be applied with some care due to the fact that 417 a potentially discontinuous function is projected onto a continuous function but it can 418 also be a desirable effect for instance to smooth a function. The projection of the 419 gradient of a function typically calculated on the element center to the 420 nodes of a FEM mesh can be evaluated on the domain boundary and so projection provides a tool to extrapolate 421 the gradient from the internal to the boundary. This is only a reasonable procedure in the absence of singularities at the boundary. 422 423 As projection is used often in simulations \escript provides an easy to use class \class{Projector} 424 which is part of the \pdetools module. The following script demonstrates 425 the usage of the class to project the piecewise constant function ($=1$ for $x\hackscore{0}\ge 0.5$ and 426 $=-1$ for $x\hackscore{0}<0.5$ ) to a function with the \ReducedSolutionFS attribute (default target) 427 \begin{python} 428 from esys.escript.pdetools import Projector 429 proj=Projector(domain) 430 x0=domain.getX() 431 jmp=1.-2.*wherePositive(x0-0.5) 432 u=proj.getValue(jmp) 433 # alternative call: 434 u=proj(jmp) 435 \end{python} 436 By default the class uses lumping to solve the PDE~\ref{PROJ.1}. This technique is faster 437 then using the standard solver techniques of PDEs. In essence it leads to using the average of 438 neighbor element values to calculate the value at each FEM node. 439 440 The following script illustrate how to evaluate the normal stress 441 on the boundary from a given displacement field \var{u}: 442 \begin{python} 443 from esys.escript.pdetools import Projector 444 u=... 445 proj=Projector(u.getDomin()) 446 e=symmetric(grad(u)) 447 stress = G*e+ (K-2./3.*G)*trace(e)*kronecker(u.getDomin()) 448 normal_stress = inner(u.getDomin().getNormal(), proj(stress)) 449 \end{python} 450 451 452 453 \begin{classdesc}{Projector}{domain\optional{, reduce=\True \optional{, fast=\True}}} 454 This class defines the projector on the domain \var{domain}. 455 If \var{reduce} is set to \True the projection will be returned 456 as a \ReducedSolutionFS \Data object. Otherwise \SolutionFS representation is returned. 457 If \var{reduce} is set to \True lumping is used when 458 the equation~\ref{PROJ.1} is solved. Otherwise the standard 459 PDE solver is used. Notice, that lumping is requires significant less 460 compute time and memory. The class is callable. 461 \end{classdesc} 462 463 \begin{methoddesc}[Projector]{getSolverOptions}{} 464 returns the solver options for solving the PDE. In fact the method returns 465 a \SolverOptions class object which can be used to modify the tolerance, 466 the solver or the preconditioner, see Section~\ref{SEC Solver Options} for details. 467 \end{methoddesc} 468 469 \begin{methoddesc}[Projector]{getValue}{input_data} 470 projects the \var{input_data}. This method is equivalent to call an instance 471 of the class with argument \var{input_data}: 472 473 \end{methoddesc} 474 475 476 % \section{Transport Problems} 477 % \label{SEC Transport} 478 479 \section{Solver Options} 480 \label{SEC Solver Options} 481 482 \begin{classdesc}{SolverOptions}{} 483 This class defines the solver options for a linear or non-linear solver. 484 The option also supports the handling of diagnostic informations. 485 \end{classdesc} 486 487 \begin{methoddesc}[SolverOptions]{getSummary}{} 488 Returns a string reporting the current settings 489 \end{methoddesc} 490 491 \begin{methoddesc}[SolverOptions]{getName}{key} 492 Returns the name as a string of a given key 493 \end{methoddesc} 494 495 \begin{methoddesc}[SolverOptions]{setSolverMethod}{\optional{method=SolverOptions.DEFAULT}} 496 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. 497 The value of \var{method} must be one of the constants 498 \member{SolverOptions.DEFAULT}, \member{SolverOptions.DIRECT}, \member{SolverOptions.CHOLEVSKY}, \member{SolverOptions.PCG},\member{SolverOptions.CR}, \member{SolverOptions.CGS}, \member{SolverOptions.BICGSTAB}, \member{SolverOptions.SSOR}, 499 \member{SolverOptions.GMRES}, \member{SolverOptions.PRES20}, \member{SolverOptions.LUMPING}, \member{SolverOptions.ITERATIVE}, \member{SolverOptions.NONLINEAR_GMRES}, \member{SolverOptions.TFQMR}, \member{SolverOptions.MINRES}, 500 or \member{SolverOptions.GAUSS_SEIDEL}. 501 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. 502 \end{methoddesc} 503 504 \begin{methoddesc}[SolverOptions]{getSolverMethod}{} 505 Returns key of the solver method to be used. 506 \end{methoddesc} 507 508 \begin{methoddesc}[SolverOptions]{setPreconditioner}{\optional{preconditioner=SolverOptions.JACOBI}} 509 Sets the preconditioner to be used. 510 The value of \var{preconditioner} must be one of the constants 511 \member{SolverOptions.ILU0}, \member{SolverOptions.ILUT}, \member{SolverOptions.JACOBI}, 512 \member{SolverOptions.AMG}, \member{SolverOptions.REC_ILU}, \member{SolverOptions.GAUSS_SEIDEL}, \member{SolverOptions.RILU}, or 513 \member{SolverOptions.NO_PRECONDITIONER}. 514 Not all packages support all preconditioner. It can be assumed that a package makes a reasonable choice if it encounters 515 an unknown preconditioner. See Table~\ref{TAB FINLEY SOLVER OPTIONS 2} for the solvers supported by \finley. 516 \end{methoddesc} 517 518 \begin{methoddesc}[SolverOptions]{getPreconditioner}{} 519 Returns key of the preconditioner to be used. 520 \end{methoddesc} 521 522 \begin{methoddesc}[SolverOptions]{setPackage}{\optional{package=SolverOptions.DEFAULT}} 523 Sets the solver package to be used as a solver. 524 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}. 525 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) 526 and, if available, \member{SolverOptions.MKL} 527 \footnote{If the stiffness matrix is non-regular \MKL may return without 528 returning a proper error code. If you observe suspicious solutions when using MKL, this may cause by a non-invertible operator. } 529 and \member{SolverOptions.UMFPACK} 530 531 \end{methoddesc} 532 533 \begin{methoddesc}[SolverOptions]{getPackage}{} 534 Returns the solver package key 535 \end{methoddesc} 536 537 538 \begin{methoddesc}[SolverOptions]{resetDiagnostics}{\optional{all=False}} 539 resets the diagnostics. If \var{all} is \True all diagnostics including accumulative counters are reset. 540 \end{methoddesc} 541 542 \begin{methoddesc}[SolverOptions]{getDiagnostics}{\optional{ name}} 543 Returns the diagnostic information \var{name}. The following keywords are 544 supported: 545 \begin{itemize} 546 \item "num_iter": the number of iteration steps 547 \item "cum_num_iter": the cumulative number of iteration steps 548 \item "num_level": the number of level in multi level solver 549 \item "num_inner_iter": the number of inner iteration steps 550 \item"cum_num_inner_iter": the cumulative number of inner iteration steps 551 \item"time": execution time 552 \item "cum_time": cumulative execution time 553 \item "set_up_time": time to set up of the solver, typically this includes factorization and reordering 554 \item "cum_set_up_time": cumulative time to set up of the solver 555 \item "net_time": net execution time, excluding setup time for the solver and execution time for preconditioner 556 \item "cum_net_time": cumulative net execution time 557 \item "residual_norm": norm of the final residual 558 \item "converged": status of convergence 559 \item "preconditioner_size": size of precondtioner in Mbytes. 560 \end{itemize} 561 \end{methoddesc} 562 563 564 \begin{methoddesc}[SolverOptions]{hasConverged}{} 565 Returns \True if the last solver call has been finalized successfully. 566 If an exception has been thrown by the solver the status of this flag is undefined. 567 \end{methoddesc} 568 569 \begin{methoddesc}[SolverOptions]{setCoarsening}{\optional{method=SolverOptions.DEFAULT}} 570 Sets the key of the coarsening method to be applied in \AMG. 571 The value of \var{method} must be one of the constants 572 \member{SolverOptions.DEFAULT} 573 \member{SolverOptions.STANDARD_COARSENING} 574 \member{SolverOptions.YAIR_SHAPIRA_COARSENING}, \\ 575 \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}. 576 \end{methoddesc} 577 578 \begin{methoddesc}[SolverOptions]{getCoarsening}{} 579 Returns the key of the coarsening algorithm to be applied \AMG. 580 \end{methoddesc} 581 582 \begin{methoddesc}[SolverOptions]{setReordering}{\optional{ordering=SolverOptions.DEFAULT_REORDERING}} 583 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 584 \member{SolverOptions.NO_REORDERING}, \member{SolverOptions.MINIMUM_FILL_IN}, 585 \member{SolverOptions.NESTED_DISSECTION}, or \member{SolverOptions.DEFAULT_REORDERING}. 586 \end{methoddesc} 587 588 \begin{methoddesc}[SolverOptions]{getReordering}{} 589 Returns the key of the reordering method to be applied if supported by the solver. 590 \end{methoddesc} 591 592 \begin{methoddesc}[SolverOptions]{setRestart}{\optional{restart=None}} 593 Sets the number of iterations steps after which \GMRES is performing a restart. 594 If \var{restart} is equal to \var{None} no restart is performed. 595 \end{methoddesc} 596 597 598 \begin{methoddesc}[SolverOptions]{getRestart}{} 599 Returns the number of iterations steps after which \GMRES is performing a restart. 600 \end{methoddesc} 601 602 \begin{methoddesc}[SolverOptions]{setTruncation}{\optional{truncation=20}} 603 Sets the number of residuals in \GMRES to be stored for orthogonalization. The more residuals are stored the faster \GMRES converged but 604 \end{methoddesc} 605 606 \begin{methoddesc}[SolverOptions]{getTruncation}{} 607 Returns the number of residuals in \GMRES to be stored for orthogonalization 608 \end{methoddesc} 609 610 611 \begin{methoddesc}[SolverOptions]{setIterMax}{\optional{iter_max=10000}} 612 Sets the maximum number of iteration steps 613 \end{methoddesc} 614 615 \begin{methoddesc}[SolverOptions]{getIterMax}{} 616 Returns maximum number of iteration steps 617 \end{methoddesc} 618 619 \begin{methoddesc}[SolverOptions]{setLevelMax}{\optional{level_max=10}} 620 Sets the maximum number of coarsening levels to be used in the \AMG solver or preconditioner. 621 \end{methoddesc} 622 623 \begin{methoddesc}[SolverOptions]{getLevelMax}{} 624 Returns the maximum number of coarsening levels to be used in an algebraic multi level solver or preconditioner 625 \end{methoddesc} 626 627 \begin{methoddesc}[SolverOptions]{setCoarseningThreshold}{\optional{theta=0.25}} 628 Sets the threshold for coarsening in the \AMG solver or preconditioner 629 \end{methoddesc} 630 631 \begin{methoddesc}[SolverOptions]{getCoarseningThreshold}{} 632 Returns the threshold for coarsening in the \AMG solver or preconditioner 633 \end{methoddesc} 634 635 \begin{methoddesc}[SolverOptions]{setMinCoarseMatrixSize}{\optional{size=500}} 636 Sets the minumum size of the coarsest level matrix in \AMG. 637 \end{methoddesc} 638 639 \begin{methoddesc}[SolverOptions]{getMinCoarseMatrixSize}{} 640 Returns the minumum size of the coarsest level matrix in \AMG. 641 \end{methoddesc} 642 643 \begin{methoddesc}[SolverOptions]{setSmoother}{\optional{smoother=\GAUSSSEIDEL}} 644 Sets the \JACOBI or \GAUSSSEIDEL smoother to be used in \AMG. 645 \end{methoddesc} 646 647 \begin{methoddesc}[SolverOptions]{getSmoother}{} 648 Returns the key for \JACOBI or \GAUSSSEIDEL smoother used in \AMG. 649 \end{methoddesc} 650 651 \begin{methoddesc}[SolverOptions]{setNumSweeps}{\optional{sweeps=2}} 652 Sets the number of sweeps in a \JACOBI or \GAUSSSEIDEL preconditioner. 653 \end{methoddesc} 654 655 \begin{methoddesc}[SolverOptions]{getNumSweeps}{} 656 Returns the number of sweeps in a \JACOBI or \GAUSSSEIDEL preconditioner. 657 \end{methoddesc} 658 659 \begin{methoddesc}[SolverOptions]{setNumPreSweeps}{\optional{sweeps=2}} 660 Sets the number of sweeps in the pre-smoothing step of \AMG 661 \end{methoddesc} 662 663 \begin{methoddesc}[SolverOptions]{getNumPreSweeps}{} 664 Returns the number of sweeps in the pre-smoothing step of \AMG 665 \end{methoddesc} 666 667 \begin{methoddesc}[SolverOptions]{setNumPostSweeps}{\optional{sweeps=2}} 668 Sets the number of sweeps in the post-smoothing step of \AMG 669 \end{methoddesc} 670 671 \begin{methoddesc}[SolverOptions]{getNumPostSweeps}{} 672 Returns he number of sweeps sweeps in the post-smoothing step of \AMG 673 \end{methoddesc} 674 675 \begin{methoddesc}[SolverOptions]{setTolerance}{\optional{rtol=1.e-8}} 676 Sets the relative tolerance for the solver. The actually meaning of tolerance depends 677 on the underlying PDE library. In most cases, the tolerance 678 will only consider the error from solving the discrete problem but will 679 not consider any discretization error. 680 \end{methoddesc} 681 682 \begin{methoddesc}[SolverOptions]{getTolerance}{} 683 Returns the relative tolerance for the solver 684 \end{methoddesc} 685 686 \begin{methoddesc}[SolverOptions]{setAbsoluteTolerance}{\optional{atol=0.}} 687 Sets the absolute tolerance for the solver. The actually meaning of tolerance depends 688 on the underlying PDE library. In most cases, the tolerance 689 will only consider the error from solving the discrete problem but will 690 not consider any discretization error. 691 \end{methoddesc} 692 693 \begin{methoddesc}[SolverOptions]{getAbsoluteTolerance}{} 694 Returns the absolute tolerance for the solver 695 \end{methoddesc} 696 697 698 \begin{methoddesc}[SolverOptions]{setInnerTolerance}{\optional{rtol=0.9}} 699 Sets the relative tolerance for an inner iteration scheme for instance 700 on the coarsest level in a multi-level scheme. 701 \end{methoddesc} 702 703 \begin{methoddesc}[SolverOptions]{getInnerTolerance}{} 704 Returns the relative tolerance for an inner iteration scheme 705 \end{methoddesc} 706 707 \begin{methoddesc}[SolverOptions]{setDropTolerance}{\optional{drop_tol=0.01}} 708 Sets the relative drop tolerance in ILUT 709 \end{methoddesc} 710 711 \begin{methoddesc}[SolverOptions]{getDropTolerance}{} 712 Returns the relative drop tolerance in \ILUT 713 \end{methoddesc} 714 715 716 \begin{methoddesc}[SolverOptions]{setDropStorage}{\optional{storage=2.}} 717 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. 718 \end{methoddesc} 719 720 \begin{methoddesc}[SolverOptions]{getDropStorage}{} 721 Returns the maximum allowed increase in storage for \ILUT 722 \end{methoddesc} 723 724 \begin{methoddesc}[SolverOptions]{setRelaxationFactor}{\optional{factor=0.3}} 725 Sets the relaxation factor used to add dropped elements in \RILU to the main diagonal. 726 \end{methoddesc} 727 728 \begin{methoddesc}[SolverOptions]{getRelaxationFactor}{} 729 Returns the relaxation factor used to add dropped elements in RILU to the main diagonal. 730 \end{methoddesc} 731 732 \begin{methoddesc}[SolverOptions]{isSymmetric}{} 733 Returns \True is the descrete system is indicated as symmetric. 734 \end{methoddesc} 735 736 \begin{methoddesc}[SolverOptions]{setSymmetryOn}{} 737 Sets the symmetry flag to indicate that the coefficient matrix is symmetric. 738 \end{methoddesc} 739 740 \begin{methoddesc}[SolverOptions]{setSymmetryOff}{} 741 Clears the symmetry flag for the coefficient matrix. 742 \end{methoddesc} 743 744 \begin{methoddesc}[SolverOptions]{isVerbose}{} 745 Returns \True if the solver is expected to be verbose. 746 \end{methoddesc} 747 748 749 \begin{methoddesc}[SolverOptions]{setVerbosityOn}{} 750 Switches the verbosity of the solver on. 751 \end{methoddesc} 752 753 754 \begin{methoddesc}[SolverOptions]{setVerbosityOff}{} 755 Switches the verbosity of the solver off. 756 \end{methoddesc} 757 758 759 \begin{methoddesc}[SolverOptions]{adaptInnerTolerance}{} 760 Returns \True if the tolerance of the inner solver is selected automatically. 761 Otherwise the inner tolerance set by \member{setInnerTolerance} is used. 762 \end{methoddesc} 763 764 \begin{methoddesc}[SolverOptions]{setInnerToleranceAdaptionOn}{} 765 Switches the automatic selection of inner tolerance on 766 \end{methoddesc} 767 768 \begin{methoddesc}[SolverOptions]{setInnerToleranceAdaptionOff}{} 769 Switches the automatic selection of inner tolerance off. 770 \end{methoddesc} 771 772 \begin{methoddesc}[SolverOptions]{setInnerIterMax}{\optional{iter_max=10}} 773 Sets the maximum number of iteration steps for the inner iteration. 774 \end{methoddesc} 775 776 \begin{methoddesc}[SolverOptions]{getInnerIterMax}{} 777 Returns maximum number of inner iteration steps. 778 \end{methoddesc} 779 780 \begin{methoddesc}[SolverOptions]{acceptConvergenceFailure}{} 781 Returns \True if a failure to meet the stopping criteria within the 782 given number of iteration steps is not raising in exception. This is useful 783 if a solver is used in a non-linear context where the non-linear solver can 784 continue even if the returned the solution does not necessarily meet the 785 stopping criteria. One can use the \member{hasConverged} method to check if the 786 last call to the solver was successful. 787 \end{methoddesc} 788 789 \begin{methoddesc}[SolverOptions]{setAcceptanceConvergenceFailureOn}{} 790 Switches the acceptance of a failure of convergence on. 791 \end{methoddesc} 792 793 \begin{methoddesc}[SolverOptions]{setAcceptanceConvergenceFailureOff}{} 794 Switches the acceptance of a failure of convergence off. 795 \end{methoddesc} 796 797 \begin{memberdesc}[SolverOptions]{DEFAULT} 798 default method, preconditioner or package to be used to solve the PDE. An appropriate method should be 799 chosen by the used PDE solver library. 800 \end{memberdesc} 801 802 \begin{memberdesc}[SolverOptions]{MKL} 803 the \MKL library by Intel,~\Ref{MKL}\footnote{The \MKL library will only be available when the Intel compilation environment is used.}. 804 \end{memberdesc} 805 806 \begin{memberdesc}[SolverOptions]{UMFPACK} 807 the \UMFPACK,~\Ref{UMFPACK}. Remark: \UMFPACK is not parallelized. 808 \end{memberdesc} 809 810 \begin{memberdesc}[SolverOptions]{PASO} 811 \PASO is the solver library of \finley, see \Sec{CHAPTER ON FINLEY}. 812 \end{memberdesc} 813 814 \begin{memberdesc}[SolverOptions]{ITERATIVE} 815 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. 816 \end{memberdesc} 817 818 \begin{memberdesc}[SolverOptions]{DIRECT} 819 the default direct linear solver. 820 \end{memberdesc} 821 822 \begin{memberdesc}[SolverOptions]{CHOLEVSKY} 823 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE. 824 \end{memberdesc} 825 826 \begin{memberdesc}[SolverOptions]{PCG} 827 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE. 828 \end{memberdesc} 829 830 \begin{memberdesc}[SolverOptions]{TFQMR} 831 transpose-free quasi-minimal residual method, see~\Ref{WEISS}\index{linear solver!TFQMR}\index{TFQMR}. \end{memberdesc} 832 833 \begin{memberdesc}[SolverOptions]{GMRES} 834 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters 835 \var{truncation} and \var{restart} of \method{getSolution}. 836 \end{memberdesc} 837 838 \begin{memberdesc}[SolverOptions]{MINRES} 839 minimal residual method method, \index{linear solver!MINRES}\index{MINRES} \end{memberdesc} 840 841 \begin{memberdesc}[SolverOptions]{LUMPING} 842 uses lumping to solve the system of linear equations~\index{linear solver!lumping}\index{lumping}. This solver technique 843 condenses the stiffness matrix to a diagonal matrix so the solution of the linear systems becomes very cheap. It can be used when 844 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 845 but is expected to converge to zero when the mesh gets finer. 846 Lumping does not use the linear system solver library. 847 \end{memberdesc} 848 849 \begin{memberdesc}[SolverOptions]{PRES20} 850 the GMRES method with truncation after five residuals and 851 restart after 20 steps, see~\Ref{WEISS}. 852 \end{memberdesc} 853 854 \begin{memberdesc}[SolverOptions]{CGS} 855 conjugate gradient squared method, see~\Ref{WEISS}. 856 \end{memberdesc} 857 858 \begin{memberdesc}[SolverOptions]{BICGSTAB} 859 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}. 860 \end{memberdesc} 861 862 \begin{memberdesc}[SolverOptions]{SSOR} 863 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support 864 this as a solver. 865 \end{memberdesc} 866 867 \begin{memberdesc}[SolverOptions]{ILU0} 868 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}. 869 \end{memberdesc} 870 871 \begin{memberdesc}[SolverOptions]{ILUT} 872 the incomplete LU factorization preconditioner with fill-in, see~\Ref{Saad}. During the LU-factorization element with 873 relative size less then \member{getDropTolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the 874 \member{getDropStorage}-fold of the stiffness matrix. \member{getDropTolerance} and \member{getDropStorage} are both set in the 875 \method{getSolution} call. 876 \end{memberdesc} 877 878 \begin{memberdesc}[SolverOptions]{JACOBI} 879 the Jacobi preconditioner, see~\Ref{Saad}. 880 \end{memberdesc} 881 882 883 \begin{memberdesc}[SolverOptions]{AMG} 884 the algebraic--multi grid method, see~\Ref{AMG}. This method can be used as linear solver method but is more robust when used 885 in a preconditioner. 886 \end{memberdesc} 887 888 \begin{memberdesc}[SolverOptions]{GAUSS_SEIDEL} 889 the symmetric Gauss-Seidel preconditioner, see~\Ref{Saad}. 890 \member{getNumSweeps()} is the number of sweeps used. 891 \end{memberdesc} 892 893 \begin{memberdesc}[SolverOptions]{RILU} 894 relaxed incomplete LU factorization preconditioner, see~\Ref{RELAXILU}. This method is similar to \ILU0 but dropped elements are added to the main diagonal 895 with the relaxation factor \member{getRelaxationFactor} 896 \end{memberdesc} 897 898 \begin{memberdesc}[SolverOptions]{REC_ILU} 899 recursive incomplete LU factorization preconditioner, see~\Ref{RILU}. This method is similar to \ILU0 but applies reordering during the factorization. 900 \end{memberdesc} 901 902 \begin{memberdesc}[SolverOptions]{NO_REORDERING} 903 no ordering is used during factorization. 904 \end{memberdesc} 905 906 \begin{memberdesc}[SolverOptions]{DEFAULT_REORDERING} 907 the default reordering method during factorization. 908 \end{memberdesc} 909 910 \begin{memberdesc}[SolverOptions]{MINIMUM_FILL_IN} 911 applies reordering before factorization using a fill-in minimization strategy. You have to check with the particular solver library or 912 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 913 \end{memberdesc} 914 915 \begin{memberdesc}[SolverOptions]{NESTED_DISSECTION} 916 applies reordering before factorization using a nested dissection strategy. You have to check with the particular solver library or 917 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 918 \end{memberdesc} 919 920 \begin{memberdesc}[SolverOptions]{TRILINOS} 921 the Trilinos library is used as a solver~\Ref{TRILINOS} 922 \end{memberdesc} 923 924 \begin{memberdesc}[SolverOptions]{SUPER_LU} 925 the SuperLU library is used as a solver~\Ref{SuperLU} 926 \end{memberdesc} 927 928 \begin{memberdesc}[SolverOptions]{PASTIX} 929 the Pastix library is used as a solver~\Ref{PASTIX} 930 \end{memberdesc} 931 932 933 \begin{memberdesc}[SolverOptions]{STANDARD_COARSENING} 934 \AMG coarsening method by Ruge and Stueben using measure of importance principle~\cite{Multigrid}. 935 \end{memberdesc} 936 937 \begin{memberdesc}[SolverOptions]{YAIR_SHAPIRA_COARSENING} 938 \AMG coarsening method by Yair-Shapira 939 \end{memberdesc} 940 941 \begin{memberdesc}[SolverOptions]{RUGE_STUEBEN_COARSENING} \AMG coarsening method by Ruge and Stueben using greedy algorithm for splitting. 942 \end{memberdesc} 943 944 \begin{memberdesc}[SolverOptions]{AGGREGATION_COARSENING} \AMG coarsening using (symmetric) aggregation using greedy algorithm for splitting. 945 \end{memberdesc} 946 947 \begin{memberdesc}[SolverOptions]{NO_PRECONDITIONER} 948 no preconditioner is applied. 949 \end{memberdesc} 950

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