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 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Copyright (c) 2003-2013 by University of Queensland 4 5 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Open Software License version 3.0 8 9 % 10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development since 2012 by School of Earth Sciences 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 21 PDE 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 $\Omega$ and $n$ 23 denotes the outer normal field on $\Gamma$. 24 25 For a single PDE with a solution that has a single component the linear PDE is 26 defined in the following form: 27 \begin{equation}\label{LINEARPDE.SINGLE.1} 28 -(A_{jl} u_{,l})_{,j}-(B_{j} u)_{,j}+C_{l} u_{,l}+D u =-X_{j,j}+Y \; . 29 \end{equation} 30 $u_{,j}$ denotes the derivative of $u$ with respect to the $j$-th spatial direction. 31 Einstein's summation convention, i.e. summation over indexes appearing twice 32 in a term of a sum, is used in this chapter. 33 The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through 34 \Data objects in the \Function on the PDE or objects that can be converted 35 into such \Data objects. 36 $A$ is a \RankTwo, $B$, $C$ and $X$ are each a \RankOne and $D$ and $Y$ are 37 scalars. 38 The following natural boundary conditions are considered\index{boundary condition!natural} on $\Gamma$: 39 \begin{equation}\label{LINEARPDE.SINGLE.2} 40 n_{j}(A_{jl} u_{,l}+B_{j} u)+d u=n_{j}X_{j} + y \;. 41 \end{equation} 42 Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. 43 The coefficients $d$ and $y$ are each a \Scalar in the \FunctionOnBoundary. 44 Constraints\index{constraint} for the solution prescribe the value of the 45 solution at certain locations in the domain. They have the form 46 \begin{equation}\label{LINEARPDE.SINGLE.3} 47 u=r \mbox{ where } q>0 48 \end{equation} 49 $r$ and $q$ are each a \Scalar where $q$ is the characteristic function\index{characteristic function} 50 defining where the constraint is applied. 51 The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other 52 condition set by \eqn{LINEARPDE.SINGLE.1} or \eqn{LINEARPDE.SINGLE.2}. 53 54 For a system of PDEs and a solution with several components the PDE has the form 55 \begin{equation}\label{LINEARPDE.SYSTEM.1} 56 -(A_{ijkl} u_{k,l})_{,j}-(B_{ijk} u_{k})_{,j}+C_{ikl} u_{k,l}+D_{ik} u_{k} =-X_{ij,j}+Y_{i} \; . 57 \end{equation} 58 $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne. 59 The natural boundary conditions\index{boundary condition!natural} take the form: 60 \begin{equation}\label{LINEARPDE.SYSTEM.2} 61 n_{j}(A_{ijkl} u_{k,l}+B_{ijk} u_{k})+d_{ik} u_{k}=n_{j}X_{ij}+y_{i} \;. 62 \end{equation} 63 The coefficient $d$ is a \RankTwo and $y$ is a \RankOne both in the 64 \FunctionOnBoundary. Constraints\index{constraint} take the form 65 \begin{equation}\label{LINEARPDE.SYSTEM.3} 66 u_{i}=r_{i} \mbox{ where } q_{i}>0 67 \end{equation} 68 $r$ and $q$ are each a \RankOne. Notice that not necessarily all components 69 must have a constraint at all locations. 70 71 \LinearPDE also supports solution discontinuities\index{discontinuity} over a 72 contact region $\Gamma^{contact}$ in the domain $\Omega$. 73 To specify the conditions across the discontinuity we are using the 74 generalised flux $J$\footnote{In some applications the definition of flux used 75 here can be different from the commonly used definition. 76 For instance, if $T$ is a temperature field the heat flux $q$ is defined as 77 $q_{,i}=-\kappa T_{,i}$ ($\kappa$ is the diffusivity) which differs from the 78 definition used here by the sign. This needs to be kept in mind when defining 79 natural boundary conditions\index{boundary condition!natural}.} which in 80 the case of a system of PDEs and several components of the solution, is 81 defined as 82 \begin{equation}\label{LINEARPDE.SYSTEM.5} 83 J_{ij}=A_{ijkl}u_{k,l}+B_{ijk}u_{k}-X_{ij} 84 \end{equation} 85 For the case of single solution component and single PDE, $J$ is defined as 86 \begin{equation}\label{LINEARPDE.SINGLE.5} 87 J_{j}=A_{jl}u_{,l}+B_{j}u_{k}-X_{j} 88 \end{equation} 89 In the context of discontinuities\index{discontinuity} $n$ denotes the normal 90 on the discontinuity pointing from side 0 towards side 1. 91 For a system of PDEs the contact condition takes the form 92 \begin{equation}\label{LINEARPDE.SYSTEM.6} 93 n_{j} J^{0}_{ij}=n_{j} J^{1}_{ij}=y^{contact}_{i} - d^{contact}_{ik} [u]_{k} \; . 94 \end{equation} 95 where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the 96 discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference 97 of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. 98 The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a 99 \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne. 100 In the case of a single PDE and a single component solution the contact 101 condition takes the form 102 \begin{equation}\label{LINEARPDE.SINGLE.6} 103 n_{j} J^{0}_{j}=n_{j} J^{1}_{j}=y^{contact} - d^{contact}[u] 104 \end{equation} 105 In this case the coefficient $d^{contact}$ and $y^{contact}$ are each a 106 \Scalar both in the \FunctionOnContactZero or \FunctionOnContactOne. 107 108 The PDE is symmetrical\index{symmetrical} if 109 \begin{equation}\label{LINEARPDE.SINGLE.4} 110 A_{jl}=A_{lj} \mbox{ and } B_{j}=C_{j} 111 \end{equation} 112 The system of PDEs is symmetrical\index{symmetrical} if 113 \begin{eqnarray} 114 \label{LINEARPDE.SYSTEM.4} 115 A_{ijkl}&=&A_{klij} \\ 116 B_{ijk}&=&C_{kij} \\ 117 D_{ik}&=&D_{ki} \\ 118 d_{ik}&=&d_{ki} \\ 119 d^{contact}_{ik}&=&d^{contact}_{ki} 120 \end{eqnarray} 121 Note that in contrast to the scalar case \eqn{LINEARPDE.SINGLE.4} now the 122 coefficients $D$, $d$ and $d^{contact}$ have to be inspected. 123 124 The following example illustrates a typical usage of the \LinearPDE class: 125 \begin{python} 126 from esys.escript import * 127 from esys.escript.linearPDEs import LinearPDE 128 from esys.finley import Rectangle 129 mydomain = Rectangle(l0=1., l1=1., n0=40, n1=20) 130 mypde=LinearPDE(mydomain) 131 mypde.setSymmetryOn() 132 mypde.setValue(A=kappa*kronecker(mydomain), D=1, Y=1) 133 u=mypde.getSolution() 134 \end{python} 135 We refer to \Chap{CHAP: Tutorial} for more details. 136 137 An instance of the \SolverOptions class is attached to the \LinearPDE class 138 object. It holds options for the solver that may be set before solving the PDE. 139 In the following example the \method{getSolverOptions} method is used to 140 access the \SolverOptions object attached to \var{mypde}: 141 \begin{python} 142 from esys.escript import * 143 from esys.escript.linearPDEs import LinearPDE, SolverOptions 144 from esys.finley import Rectangle 145 mydomain = Rectangle(l0=1., l1=1., n0=40, n1=20) 146 mypde=LinearPDE(mydomain) 147 mypde.setValue(A=kappa*kronecker(mydomain), D=1, Y=1) 148 mypde.getSolverOptions().setVerbosityOn() 149 mypde.getSolverOptions().setSolverMethod(SolverOptions.PCG) 150 mypde.getSolverOptions().setPreconditioner(SolverOptions.AMG) 151 mypde.getSolverOptions().setTolerance(1e-8) 152 mypde.getSolverOptions().setIterMax(1000) 153 u=mypde.getSolution() 154 \end{python} 155 In this example, the preconditioned conjugate gradient method \PCG is used 156 with preconditioner \AMG. The relative tolerance is set to $10^{-8}$ and 157 the maximum number of iteration steps to $1000$. 158 After a completed call to \method{getSolution()}, the attached \SolverOptions 159 object gives access to diagnostic information: 160 \begin{python} 161 u=mypde.getSolution() 162 print("Number of iteration steps =", mypde.getDiagnostics("num_iter")) 163 print("Total solution time =", mypde.getDiagnostics("time")) 164 print("Set-up time =", mypde.getDiagnostics("set_up_time")) 165 print("Net time =", mypde.getDiagnostics("net_time")) 166 print("Residual norm of returned solution =", mypde.getDiagnostics('residual_norm')) 167 \end{python} 168 Typically, a negative value for a diagnostic variable indicates that it is 169 undefined. 170 171 \subsection{Classes} 172 %\declaremodule{extension}{esys.escript.linearPDEs} 173 %\modulesynopsis{Linear partial differential equation handler} 174 The module \linearPDEs provides an interface to define and solve linear partial 175 differential equations within \escript. The module \linearPDEs does not 176 provide any solver capabilities in itself but hands the PDE over to the PDE 177 solver library defined through the \Domain of the PDE, e.g. \finley. 178 The general interface is provided through the \LinearPDE class. The \Poisson 179 class which is also derived form the \LinearPDE class should be used to define 180 the Poisson equation\index{Poisson}. 181 182 \subsection{\LinearPDE class} 183 This is the general class to define a linear PDE in \escript. 184 We list a selection of the most important methods of the class. 185 For a complete list, see the reference at \ReferenceGuide. 186 187 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0} 188 opens a linear, steady, second order PDE on the \Domain \var{domain}. 189 The parameters \var{numEquations} and \var{numSolutions} give the number of 190 equations and the number of solution components. 191 If \var{numEquations} and \var{numSolutions} are non-positive, then the number 192 of equations and the number of solutions, respectively, stay undefined until a 193 coefficient is defined. 194 \end{classdesc} 195 196 \subsubsection{\LinearPDE methods} 197 \begin{methoddesc}[LinearPDE]{setValue}{ 198 \optional{A}\optional{, B}, 199 \optional{, C}\optional{, D} 200 \optional{, X}\optional{, Y} 201 \optional{, d}\optional{, y} 202 \optional{, d_contact}\optional{, y_contact} 203 \optional{, q}\optional{, r}} 204 assigns new values to coefficients. By default all values are assumed to be 205 zero\footnote{In fact, it is assumed they are not present by assigning the 206 value \code{escript.Data()}. This can be used by the solver library to reduce 207 computational costs.}. 208 If the new coefficient value is not a \Data object, it is converted into a 209 \Data object in the appropriate \FunctionSpace. 210 \end{methoddesc} 211 212 \begin{methoddesc}[LinearPDE]{getCoefficient}{name} 213 returns the value assigned to coefficient \var{name}. If \var{name} is not a 214 valid name an exception is raised. 215 \end{methoddesc} 216 217 \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name} 218 returns the shape of the coefficient \var{name} even if no value has been 219 assigned to it. 220 \end{methoddesc} 221 222 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name} 223 returns the \FunctionSpace of the coefficient \var{name} even if no value has 224 been assigned to it. 225 \end{methoddesc} 226 227 \begin{methoddesc}[LinearPDE]{setDebugOn}{} 228 switches on debug mode so more diagnostic messages will be printed. 229 \end{methoddesc} 230 231 \begin{methoddesc}[LinearPDE]{setDebugOff}{} 232 switches off debug mode. 233 \end{methoddesc} 234 235 \begin{methoddesc}[LinearPDE]{getSolverOptions}{} 236 returns the solver options for solving the PDE. In fact, the method returns 237 a \SolverOptions class object which can be used to modify the tolerance, 238 the solver or the preconditioner, see \Sec{SEC Solver Options} for details. 239 \end{methoddesc} 240 241 \begin{methoddesc}[LinearPDE]{setSolverOptions}{\optional{options=None}} 242 sets the solver options for solving the PDE. If argument \var{options} is 243 present it must be a \SolverOptions class object, see \Sec{SEC Solver Options} 244 for details. Otherwise the solver options are reset to the default. 245 \end{methoddesc} 246 247 \begin{methoddesc}[LinearPDE]{isUsingLumping}{} 248 returns \True if matrix lumping is set as the solver for the system of linear 249 equations, \False otherwise. 250 \end{methoddesc} 251 252 \begin{methoddesc}[LinearPDE]{getDomain}{} 253 returns the \Domain of the PDE. 254 \end{methoddesc} 255 256 \begin{methoddesc}[LinearPDE]{getDim}{} 257 returns the spatial dimension of the PDE. 258 \end{methoddesc} 259 260 \begin{methoddesc}[LinearPDE]{getNumEquations}{} 261 returns the number of equations. 262 \end{methoddesc} 263 264 \begin{methoddesc}[LinearPDE]{getNumSolutions}{} 265 returns the number of components of the solution. 266 \end{methoddesc} 267 268 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False} 269 returns \True if the PDE is symmetric, \False otherwise. 270 The method is very computationally expensive and should only be called for 271 testing purposes. The symmetry flag is not altered. 272 If \var{verbose=True} information about where symmetry is violated is printed. 273 \end{methoddesc} 274 275 \begin{methoddesc}[LinearPDE]{getFlux}{u} 276 returns the flux $J_{ij}$\index{flux} for given solution \var{u} defined by 277 \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}. 278 \end{methoddesc} 279 280 \begin{methoddesc}[LinearPDE]{isSymmetric}{} 281 returns \True if the PDE has been indicated to be symmetric, \False otherwise. 282 \end{methoddesc} 283 284 \begin{methoddesc}[LinearPDE]{setSymmetryOn}{} 285 indicates that the PDE is symmetric. 286 \end{methoddesc} 287 288 \begin{methoddesc}[LinearPDE]{setSymmetryOff}{} 289 indicates that the PDE is not symmetric. 290 \end{methoddesc} 291 292 \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{} 293 enables the reduction of polynomial order for the solution and equation 294 evaluation even if a quadratic or higher interpolation order is defined in the 295 \Domain. This feature may not be supported by all PDE libraries. 296 \end{methoddesc} 297 298 \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{} 299 disables the reduction of polynomial order for the solution and equation evaluation. 300 \end{methoddesc} 301 302 \begin{methoddesc}[LinearPDE]{getOperator}{} 303 returns the \Operator of the PDE. 304 \end{methoddesc} 305 306 \begin{methoddesc}[LinearPDE]{getRightHandSide}{} 307 returns the right hand side of the PDE as a \Data object. 308 If \var{ignoreConstraint=True}, then the constraints are not considered when 309 building up the right hand side. 310 \end{methoddesc} 311 312 \begin{methoddesc}[LinearPDE]{getSystem}{} 313 returns the \Operator and right hand side of the PDE. 314 \end{methoddesc} 315 316 \begin{methoddesc}[LinearPDE]{getSolution}{} 317 returns (an approximation of) the solution of the PDE. This call will invoke 318 the discretization of the PDE and the solution of the resulting system of 319 linear equations. Keep in mind that this call is typically computationally 320 expensive and -- depending on the PDE and the discretization -- can take a 321 long time to complete. 322 \end{methoddesc} 323 324 \subsection{The \Poisson Class} 325 The \Poisson class provides an easy way to define and solve the Poisson 326 equation 327 \begin{equation}\label{POISSON.1} 328 -u_{,ii}=f 329 \end{equation} 330 with homogeneous boundary conditions 331 \begin{equation}\label{POISSON.2} 332 n_{i}u_{,i}=0 333 \end{equation} 334 and homogeneous constraints 335 \begin{equation}\label{POISSON.3} 336 u=0 \mbox{ where } q>0 . 337 \end{equation} 338 $f$ has to be a \Scalar in the \Function and $q$ must be a \Scalar in the \SolutionFS. 339 340 \begin{classdesc}{Poisson}{domain} 341 opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE. 342 \end{classdesc} 343 \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()} 344 assigns new values to \var{f} and \var{q}. 345 \end{methoddesc} 346 347 \subsection{The \Helmholtz Class} 348 The \Helmholtz class defines the Helmholtz problem 349 \begin{equation}\label{HZ.1} 350 \omega \; u - (k\; u_{,j})_{,j} = f 351 \end{equation} 352 with natural boundary conditions 353 \begin{equation}\label{HZ.2} 354 k\; u_{,j} n_{,j} = g- \alpha \; u 355 \end{equation} 356 and constraints 357 \begin{equation}\label{HZ.3} 358 u=r \mbox{ where } q>0 . 359 \end{equation} 360 $\omega$, $k$, and $f$ each have to be a \Scalar in the \Function, $g$ and 361 $\alpha$ must be a \Scalar in the \FunctionOnBoundary, and $q$ and $r$ must be 362 a \Scalar in the \SolutionFS or must be mapped or interpolated into the 363 particular \FunctionSpace. 364 365 \begin{classdesc}{Helmholtz}{domain} 366 opens a Helmholtz equation on the \Domain domain. \Helmholtz is derived from \LinearPDE. 367 \end{classdesc} 368 \begin{methoddesc}[Helmholtz]{setValue}{ \optional{omega} \optional{, k} \optional{, f} \optional{, alpha} \optional{, g} \optional{, r} \optional{, q}} 369 assigns new values to \var{omega}, \var{k}, \var{f}, \var{alpha}, \var{g}, 370 \var{r}, and \var{q}. By default all values are set to zero. 371 \end{methoddesc} 372 373 \subsection{The \Lame Class} 374 The \Lame class defines a Lame equation problem 375 \begin{equation}\label{LE.1} 376 -(\mu (u_{i,j}+u_{j,i})+\lambda u_{k,k}\delta_{ij})_{j} = F_{i}-\sigma_{ij,j} 377 \end{equation} 378 with natural boundary conditions 379 \begin{equation}\label{LE.2} 380 n_{j}(\mu \; (u_{i,j}+u_{j,i})+\lambda u_{k,k}\delta_{ij}) = f_{i}+n_{j}\sigma_{ij} 381 \end{equation} 382 and constraint 383 \begin{equation}\label{LE.3} 384 u_{i}=r_{i} \mbox{ where } q_{i}>0 . 385 \end{equation} 386 $\mu$, $\lambda$ have to be a \Scalar in the \Function, $F$ has to be a 387 \Vector in the \Function, $\sigma$ has to be a \Tensor in the \Function, 388 $f$ must be a \Vector in the \FunctionOnBoundary, and $q$ and $r$ must be a 389 \Vector in the \SolutionFS or must be mapped or interpolated into the 390 particular \FunctionSpace. 391 392 \begin{classdesc}{Lame}{domain} 393 opens a Lame equation on the \Domain domain. \Lame is derived from \LinearPDE. 394 \end{classdesc} 395 \begin{methoddesc}[Lame]{setValue}{ \optional{lame_lambda} \optional{, lame_mu} \optional{, F} \optional{, sigma} \optional{, f} \optional{, r} \optional{, q}} 396 assigns new values to \var{lame_lambda}, \var{lame_mu}, \var{F}, \var{sigma}, 397 \var{f}, \var{r}, and \var{q}. By default all values are set to zero. 398 \end{methoddesc} 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 405 attribute in addition to the standard interpolation mechanism\index{interpolation} 406 as discussed in \Chap{ESCRIPT CHAP}. If you consider the stripped-down version 407 \begin{equation}\label{PROJ.1} 408 u = Y 409 \end{equation} 410 of the general scalar PDE~\ref{LINEARPDE.SINGLE.1} (boundary conditions are 411 irrelevant), you can see the solution $u$ of this PDE as a projection of the 412 input function $Y$ which has the \Function attribute to a function with the 413 \SolutionFS or \ReducedSolutionFS attribute. 414 In fact, the solution maps values defined at element centers representing a 415 possibly discontinuous function onto a continuous function represented by its 416 values at the nodes of the FEM mesh. 417 This mapping is called a projection\index{projection}. Projection can be a 418 useful tool but needs to be applied with some care due to the possibility of 419 projecting a potentially discontinuous function onto a continuous function, 420 although this may also be a desirable effect, for instance to smooth a function. 421 The projection of the gradient of a function typically calculated on the 422 element center to the nodes of a FEM mesh can be evaluated on the domain 423 boundary and so projection provides a tool to extrapolate the gradient from 424 the internal to the boundary. This is only a reasonable procedure in the 425 absence of singularities at the boundary. 426 427 As projection is often used in simulations \escript provides an easy to use 428 class \class{Projector} which is part of the \pdetools module. 429 The following script demonstrates the usage of the class to project the 430 piecewise constant function ($=1$ for $x_{0}\ge 0.5$ and $=-1$ for $x_{0}<0.5$) 431 to a function with the \ReducedSolutionFS attribute (default target): 432 \begin{python} 433 from esys.escript.pdetools import Projector 434 proj=Projector(domain) 435 x0=domain.getX() 436 jmp=1.-2.*wherePositive(x0-0.5) 437 u=proj.getValue(jmp) 438 # alternative call: 439 u=proj(jmp) 440 \end{python} 441 By default the class uses lumping to solve the PDE~\ref{PROJ.1}. 442 This technique is faster than using the standard solver techniques of PDEs. 443 In essence it leads to using the average of neighbour element values to 444 calculate the value at each FEM node. 445 446 The following script illustrates how to evaluate the normal stress on the 447 boundary from a given displacement field \var{u}: 448 \begin{python} 449 from esys.escript.pdetools import Projector 450 u=... 451 proj=Projector(u.getDomain()) 452 e=symmetric(grad(u)) 453 stress = G*e+ (K-2./3.*G)*trace(e)*kronecker(u.getDomain()) 454 normal_stress = inner(u.getDomain().getNormal(), proj(stress)) 455 \end{python} 456 457 \begin{classdesc}{Projector}{domain\optional{, reduce=\True \optional{, fast=\True}}} 458 This class defines a projector on the domain \var{domain}. 459 If \var{reduce} is set to \True the projection will be returned as a 460 \ReducedSolutionFS \Data object. 461 Otherwise the \SolutionFS representation is returned. 462 If \var{reduce} is set to \True lumping is used when the \eqn{PROJ.1} is 463 solved, otherwise the standard PDE solver is used. 464 Notice, that lumping requires significantly less computation time and memory. 465 The class is callable. 466 \end{classdesc} 467 468 \begin{methoddesc}[Projector]{getSolverOptions}{} 469 returns the solver options for solving the PDE. In fact, the method returns 470 a \SolverOptions class object which can be used to modify the tolerance, 471 the solver or the preconditioner, see \Sec{SEC Solver Options} for details. 472 \end{methoddesc} 473 474 \begin{methoddesc}[Projector]{getValue}{input_data} 475 projects the \var{input_data}. This method is equivalent to call an instance 476 of the class with argument \var{input_data} 477 \end{methoddesc} 478 479 % \section{Transport Problems} 480 % \label{SEC Transport} 481 482 \section{Solver Options} 483 \label{SEC Solver Options} 484 485 \begin{classdesc}{SolverOptions}{} 486 This class defines the solver options for a linear or non-linear solver. 487 The option also supports the handling of diagnostic information. 488 \end{classdesc} 489 490 \begin{methoddesc}[SolverOptions]{getSummary}{} 491 returns a string reporting the current settings. 492 \end{methoddesc} 493 494 \begin{methoddesc}[SolverOptions]{getName}{key} 495 returns the name as a string of a given key. 496 \end{methoddesc} 497 498 \begin{methoddesc}[SolverOptions]{setSolverMethod}{\optional{method=SolverOptions.DEFAULT}} 499 sets the solver method to be used. 500 Use \var{method}=\member{SolverOptions.DIRECT} to indicate that a direct 501 rather than an iterative solver should be used and use 502 \var{method}=\member{SolverOptions.ITERATIVE} to indicate that an iterative 503 rather than a direct solver should be used. 504 The value of \var{method} must be one of the constants:\\ 505 \member{SolverOptions.DEFAULT}\\ 506 \member{SolverOptions.DIRECT}\\ 507 \member{SolverOptions.CHOLEVSKY}\\ 508 \member{SolverOptions.PCG}\\ 509 \member{SolverOptions.CR}\\ 510 \member{SolverOptions.CGS}\\ 511 \member{SolverOptions.BICGSTAB}\\ 512 \member{SolverOptions.SSOR}\\ 513 \member{SolverOptions.GMRES}\\ 514 \member{SolverOptions.PRES20}\\ 515 \member{SolverOptions.ROWSUM_LUMPING}\\ 516 \member{SolverOptions.HRZ_LUMPING}\\ 517 \member{SolverOptions.ITERATIVE}\\ 518 \member{SolverOptions.NONLINEAR_GMRES}\\ 519 \member{SolverOptions.TFQMR}\\ 520 \member{SolverOptions.MINRES}\\ 521 \member{SolverOptions.GAUSS_SEIDEL}.\\ 522 Not all packages support all solvers. It can be assumed that a package makes a 523 reasonable choice if it encounters an unknown solver. 524 See Table~\ref{TAB FINLEY SOLVER OPTIONS 1} for the solvers supported by 525 \finley. 526 \end{methoddesc} 527 528 \begin{methoddesc}[SolverOptions]{getSolverMethod}{} 529 returns the key of the solver method to be used. 530 \end{methoddesc} 531 532 \begin{methoddesc}[SolverOptions]{setPreconditioner}{\optional{preconditioner=SolverOptions.JACOBI}} 533 sets the preconditioner to be used. 534 The value of \var{preconditioner} must be one of the constants:\\ 535 \member{SolverOptions.ILU0}\\ 536 \member{SolverOptions.JACOBI}\\ 537 \member{SolverOptions.AMG}\\ 538 \member{SolverOptions.REC_ILU}\\ 539 \member{SolverOptions.GAUSS_SEIDEL}\\ 540 \member{SolverOptions.RILU}\\ 541 \member{SolverOptions.NO_PRECONDITIONER}.\\ 542 Not all packages support all preconditioners. It can be assumed that a package 543 makes a reasonable choice if it encounters an unknown preconditioner. 544 See Table~\ref{TAB FINLEY SOLVER OPTIONS 2} for the preconditioners supported 545 by \finley. 546 \end{methoddesc} 547 548 \begin{methoddesc}[SolverOptions]{getPreconditioner}{} 549 returns the key of the preconditioner to be used. 550 \end{methoddesc} 551 552 \begin{methoddesc}[SolverOptions]{setPackage}{\optional{package=SolverOptions.DEFAULT}} 553 sets the solver package to be used as a solver. 554 The value of \var{method} must be one of the constants:\\ 555 \member{SolverOptions.DEFAULT}\\ 556 \member{SolverOptions.PASO}\\ 557 \member{SolverOptions.SUPER_LU}\\ 558 \member{SolverOptions.PASTIX}\\ 559 \member{SolverOptions.MKL}\\ 560 \member{SolverOptions.UMFPACK}.\\ 561 Not all packages are supported on all implementations. An exception may be 562 thrown on some platforms if a particular package is requested. 563 Currently \finley supports \member{SolverOptions.PASO} (as default) and, if 564 available, \member{SolverOptions.MKL}\footnote{If the stiffness matrix is 565 non-regular \MKL may return without returning a proper error code. 566 If you observe suspicious solutions when using MKL, this may be causes by a 567 non-invertible operator.} and \member{SolverOptions.UMFPACK}. 568 \end{methoddesc} 569 570 \begin{methoddesc}[SolverOptions]{getPackage}{} 571 returns the solver package key. 572 \end{methoddesc} 573 574 \begin{methoddesc}[SolverOptions]{resetDiagnostics}{\optional{all=False}} 575 resets the diagnostics. If \var{all} is \True all diagnostics, including 576 accumulative counters, are reset. 577 \end{methoddesc} 578 579 \begin{methoddesc}[SolverOptions]{getDiagnostics}{\optional{ name}} 580 returns the diagnostic information \var{name}. The following keywords are 581 supported:\\ 582 \var{"num_iter"}: number of iteration steps\\ 583 \var{"cum_num_iter"}: cumulative number of iteration steps\\ 584 \var{"num_level"}: number of levels in the multi level solver\\ 585 \var{"num_inner_iter"}: number of inner iteration steps\\ 586 \var{"cum_num_inner_iter"}: cumulative number of inner iteration steps\\ 587 \var{"time"}: execution time\\ 588 \var{"cum_time"}: cumulative execution time\\ 589 \var{"set_up_time"}: time to set up the solver, typically this includes 590 factorization and reordering\\ 591 \var{"cum_set_up_time"}: cumulative time to set up the solver\\ 592 \var{"net_time"}: net execution time, excluding setup time for the solver 593 and execution time for preconditioner\\ 594 \var{"cum_net_time"}: cumulative net execution time\\ 595 \var{"residual_norm"}: norm of the final residual\\ 596 \var{"converged"}: status of convergence\\ 597 \var{"preconditioner_size"}: size of preconditioner in MBytes \\ 598 \var{"preconditioner_size"}: size of preconditioner in MBytes \\ 599 \var{"preconditioner_size"}: size of preconditioner in MBytes . 600 601 \end{methoddesc} 602 603 \begin{methoddesc}[SolverOptions]{hasConverged}{} 604 returns \True if the last solver call has been finalized successfully. 605 If an exception has been thrown by the solver the status of this flag is undefined. 606 \end{methoddesc} 607 608 609 \begin{methoddesc}[SolverOptions]{setReordering}{\optional{ordering=SolverOptions.DEFAULT_REORDERING}} 610 sets the key of the reordering method to be applied if supported by the solver. 611 Some direct solvers support reordering to optimize compute time and storage 612 use during elimination. The value of \var{ordering} must be one of the 613 constants:\\ 614 \member{SolverOptions.NO_REORDERING}\\ 615 \member{SolverOptions.MINIMUM_FILL_IN}\\ 616 \member{SolverOptions.NESTED_DISSECTION}\\ 617 \member{SolverOptions.DEFAULT_REORDERING}. 618 \end{methoddesc} 619 620 \begin{methoddesc}[SolverOptions]{getReordering}{} 621 returns the key of the reordering method to be applied if supported by the solver. 622 \end{methoddesc} 623 624 \begin{methoddesc}[SolverOptions]{setRestart}{\optional{restart=None}} 625 sets the number of iterations steps after which \GMRES is to perform a restart. 626 If \var{restart} is equal to \var{None} no restart is performed. 627 \end{methoddesc} 628 629 \begin{methoddesc}[SolverOptions]{getRestart}{} 630 returns the number of iterations steps after which \GMRES performs a restart. 631 \end{methoddesc} 632 633 \begin{methoddesc}[SolverOptions]{setTruncation}{\optional{truncation=20}} 634 sets the number of residuals in \GMRES to be stored for orthogonalization. 635 The more residuals are stored the faster \GMRES converges but the higher the storage needs are and the more expensive 636 a single iteration step becomes. 637 \end{methoddesc} 638 639 \begin{methoddesc}[SolverOptions]{getTruncation}{} 640 returns the number of residuals in \GMRES to be stored for orthogonalization. 641 \end{methoddesc} 642 643 644 \begin{methoddesc}[SolverOptions]{setIterMax}{\optional{iter_max=10000}} 645 sets the maximum number of iteration steps. 646 \end{methoddesc} 647 648 \begin{methoddesc}[SolverOptions]{getIterMax}{} 649 returns maximum number of iteration steps. 650 \end{methoddesc} 651 652 \begin{methoddesc}[SolverOptions]{setLevelMax}{\optional{level_max=10}} 653 sets the maximum number of coarsening levels to be used in the \AMG solver or 654 preconditioner. 655 \end{methoddesc} 656 657 \begin{methoddesc}[SolverOptions]{getLevelMax}{} 658 returns the maximum number of coarsening levels to be used in an algebraic 659 multi level solver or preconditioner. 660 \end{methoddesc} 661 662 \begin{methoddesc}[SolverOptions]{setCoarseningThreshold}{\optional{theta=0.25}} 663 sets the threshold for coarsening in the \AMG solver or preconditioner. 664 \end{methoddesc} 665 666 \begin{methoddesc}[SolverOptions]{getCoarseningThreshold}{} 667 returns the threshold for coarsening in the \AMG solver or preconditioner. 668 \end{methoddesc} 669 670 \begin{methoddesc}[SolverOptions]{setDiagonalDominanceThreshold}{\optional{value=0.5}} 671 sets the threshold for diagonal dominant rows which are eliminated during \AMG coarsening. 672 \end{methoddesc} 673 674 \begin{methoddesc}[SolverOptions]{getDiagonalDominanceThreshold}{} 675 returns the threshold for diagonal dominant rows which are eliminated during \AMG coarsening. 676 \end{methoddesc} 677 678 \begin{methoddesc}[SolverOptions]{setMinCoarseMatrixSize}{\optional{size=500}} 679 sets the minimum size of the coarsest level matrix in \AMG. 680 \end{methoddesc} 681 682 \begin{methoddesc}[SolverOptions]{getMinCoarseMatrixSize}{} 683 returns the minimum size of the coarsest level matrix in \AMG. 684 \end{methoddesc} 685 686 \begin{methoddesc}[SolverOptions]{setSmoother}{\optional{smoother=\GAUSSSEIDEL}} 687 sets the \JACOBI or \GAUSSSEIDEL smoother to be used with \AMG. 688 \end{methoddesc} 689 690 \begin{methoddesc}[SolverOptions]{getSmoother}{} 691 returns the key of the smoother used in \AMG. 692 \end{methoddesc} 693 694 \begin{methoddesc}[SolverOptions]{setAMGInterpolation}{\optional{method=\var{None}}} 695 sets interpolation method for \AMG to 696 \member{CLASSIC_INTERPOLATION_WITH_FF_COUPLING}, 697 \member{CLASSIC_INTERPOLATION}, or 698 \member{DIRECT_INTERPOLATION}. 699 \end{methoddesc} 700 701 \begin{methoddesc}[SolverOptions]{getAMGInterpolation}{} 702 returns the key 703 \member{CLASSIC_INTERPOLATION_WITH_FF_COUPLING}, 704 \member{CLASSIC_INTERPOLATION}, or 705 \member{DIRECT_INTERPOLATION} 706 of the interpolation method for \AMG. 707 \end{methoddesc} 708 709 \begin{methoddesc}[SolverOptions]{setNumSweeps}{\optional{sweeps=2}} 710 sets the number of sweeps in a \JACOBI or \GAUSSSEIDEL preconditioner. 711 \end{methoddesc} 712 713 \begin{methoddesc}[SolverOptions]{getNumSweeps}{} 714 returns the number of sweeps in a \JACOBI or \GAUSSSEIDEL preconditioner. 715 \end{methoddesc} 716 717 718 719 \begin{methoddesc}[SolverOptions]{setNumPreSweeps}{\optional{sweeps=2}} 720 sets the number of sweeps in the pre-smoothing step of \AMG. 721 \end{methoddesc} 722 723 \begin{methoddesc}[SolverOptions]{getNumPreSweeps}{} 724 returns the number of sweeps in the pre-smoothing step of \AMG. 725 \end{methoddesc} 726 727 \begin{methoddesc}[SolverOptions]{setNumPostSweeps}{\optional{sweeps=2}} 728 sets the number of sweeps in the post-smoothing step of \AMG. 729 \end{methoddesc} 730 731 \begin{methoddesc}[SolverOptions]{getNumPostSweeps}{} 732 returns he number of sweeps in the post-smoothing step of \AMG. 733 \end{methoddesc} 734 735 \begin{methoddesc}[SolverOptions]{setTolerance}{\optional{rtol=1.e-8}} 736 sets the relative tolerance for the solver. The actual meaning of tolerance 737 depends on the underlying PDE library. In most cases, the tolerance 738 will only consider the error from solving the discrete problem but will 739 not consider any discretization error. 740 \end{methoddesc} 741 742 \begin{methoddesc}[SolverOptions]{getTolerance}{} 743 returns the relative tolerance for the solver. 744 \end{methoddesc} 745 746 \begin{methoddesc}[SolverOptions]{setAbsoluteTolerance}{\optional{atol=0.}} 747 sets the absolute tolerance for the solver. The actual meaning of tolerance 748 depends on the underlying PDE library. In most cases, the tolerance 749 will only consider the error from solving the discrete problem but will 750 not consider any discretization error. 751 \end{methoddesc} 752 753 \begin{methoddesc}[SolverOptions]{getAbsoluteTolerance}{} 754 returns the absolute tolerance for the solver. 755 \end{methoddesc} 756 757 \begin{methoddesc}[SolverOptions]{setInnerTolerance}{\optional{rtol=0.9}} 758 sets the relative tolerance for an inner iteration scheme, for instance 759 on the coarsest level in a multi-level scheme. 760 \end{methoddesc} 761 762 \begin{methoddesc}[SolverOptions]{getInnerTolerance}{} 763 returns the relative tolerance for an inner iteration scheme. 764 \end{methoddesc} 765 766 767 \begin{methoddesc}[SolverOptions]{setRelaxationFactor}{\optional{factor=0.3}} 768 sets the relaxation factor used to add dropped elements in \RILU to the main diagonal. 769 \end{methoddesc} 770 771 \begin{methoddesc}[SolverOptions]{getRelaxationFactor}{} 772 returns the relaxation factor used to add dropped elements in \RILU to the main diagonal. 773 \end{methoddesc} 774 775 \begin{methoddesc}[SolverOptions]{isSymmetric}{} 776 returns \True if the discrete system is indicated as symmetric. 777 \end{methoddesc} 778 779 \begin{methoddesc}[SolverOptions]{setSymmetryOn}{} 780 sets the symmetry flag to indicate that the coefficient matrix is symmetric. 781 \end{methoddesc} 782 783 \begin{methoddesc}[SolverOptions]{setSymmetryOff}{} 784 clears the symmetry flag for the coefficient matrix. 785 \end{methoddesc} 786 787 \begin{methoddesc}[SolverOptions]{isVerbose}{} 788 returns \True if the solver is expected to be verbose. 789 \end{methoddesc} 790 791 \begin{methoddesc}[SolverOptions]{setVerbosityOn}{} 792 switches the verbosity of the solver on. 793 \end{methoddesc} 794 795 \begin{methoddesc}[SolverOptions]{setVerbosityOff}{} 796 switches the verbosity of the solver off. 797 \end{methoddesc} 798 799 \begin{methoddesc}[SolverOptions]{adaptInnerTolerance}{} 800 returns \True if the tolerance of the inner solver is selected automatically. 801 Otherwise the inner tolerance set by \member{setInnerTolerance} is used. 802 \end{methoddesc} 803 804 \begin{methoddesc}[SolverOptions]{setInnerToleranceAdaptionOn}{} 805 switches the automatic selection of inner tolerance on. 806 \end{methoddesc} 807 808 \begin{methoddesc}[SolverOptions]{setInnerToleranceAdaptionOff}{} 809 switches the automatic selection of inner tolerance off. 810 \end{methoddesc} 811 812 \begin{methoddesc}[SolverOptions]{setInnerIterMax}{\optional{iter_max=10}} 813 sets the maximum number of iteration steps for the inner iteration. 814 \end{methoddesc} 815 816 \begin{methoddesc}[SolverOptions]{getInnerIterMax}{} 817 returns the maximum number of inner iteration steps. 818 \end{methoddesc} 819 820 \begin{methoddesc}[SolverOptions]{acceptConvergenceFailure}{} 821 returns \True if a failure to meet the stopping criteria within the given 822 number of iteration steps is not raising in exception. This is useful 823 if a solver is used in a non-linear context where the non-linear solver can 824 continue even if the returned solution does not necessarily meet the stopping 825 criteria. One can use the \member{hasConverged} method to check if the 826 last call to the solver was successful. 827 \end{methoddesc} 828 829 \begin{methoddesc}[SolverOptions]{setAcceptanceConvergenceFailureOn}{} 830 switches the acceptance of a failure of convergence on. 831 \end{methoddesc} 832 833 \begin{methoddesc}[SolverOptions]{setAcceptanceConvergenceFailureOff}{} 834 switches the acceptance of a failure of convergence off. 835 \end{methoddesc} 836 837 \begin{memberdesc}[SolverOptions]{DEFAULT} 838 default method, preconditioner or package to be used to solve the PDE. 839 An appropriate method should be chosen by the used PDE solver library. 840 \end{memberdesc} 841 842 \begin{memberdesc}[SolverOptions]{MKL} 843 the \MKL library by Intel, \Ref{MKL}\footnote{The \MKL library will only be 844 available when the Intel compilation environment was used to build \escript.}. 845 \end{memberdesc} 846 847 \begin{memberdesc}[SolverOptions]{UMFPACK} 848 the \UMFPACK library, \Ref{UMFPACK}. Note that \UMFPACK is not parallelized. 849 \end{memberdesc} 850 851 \begin{memberdesc}[SolverOptions]{PASO} 852 \PASO is the default solver library of \finley, see \Sec{CHAPTER ON FINLEY}. 853 \end{memberdesc} 854 855 \begin{memberdesc}[SolverOptions]{ITERATIVE} 856 the default iterative method and preconditioner. The actual method used 857 depends on the PDE solver library and the chosen solver package. 858 Typically, \PCG is used for symmetric PDEs and \BiCGStab otherwise, both with 859 \JACOBI preconditioner. 860 \end{memberdesc} 861 862 \begin{memberdesc}[SolverOptions]{DIRECT} 863 the default direct linear solver. 864 \end{memberdesc} 865 866 \begin{memberdesc}[SolverOptions]{CHOLEVSKY} 867 direct solver based on Cholevsky factorization (or similar), see \Ref{Saad}. 868 The solver requires a symmetric PDE. 869 \end{memberdesc} 870 871 \begin{memberdesc}[SolverOptions]{PCG} 872 preconditioned conjugate gradient method, see \Ref{WEISS}\index{linear solver!PCG}\index{PCG}. 873 The solver requires a symmetric PDE. 874 \end{memberdesc} 875 876 \begin{memberdesc}[SolverOptions]{TFQMR} 877 transpose-free quasi-minimal residual method, see \Ref{WEISS}\index{linear solver!TFQMR}\index{TFQMR}. 878 \end{memberdesc} 879 880 \begin{memberdesc}[SolverOptions]{GMRES} 881 the GMRES method, see \Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. 882 Truncation and restart are controlled by the \var{truncation} and \var{restart} 883 parameters of \method{getSolution}. 884 \end{memberdesc} 885 886 \begin{memberdesc}[SolverOptions]{MINRES} 887 minimal residual method\index{linear solver!MINRES}\index{MINRES} 888 \end{memberdesc} 889 890 \begin{memberdesc}[SolverOptions]{ROWSUM_LUMPING} 891 row sum lumping of the stiffness matrix, see Section~\ref{LUMPING} for details\index{linear solver!row sum lumping}\index{row sum lumping}. 892 Lumping does not use the linear system solver library. 893 \end{memberdesc} 894 895 \begin{memberdesc}[SolverOptions]{HRZ_LUMPING} 896 HRZ lumping of the stiffness matrix, see Section~\ref{LUMPING} for details\index{linear solver!HRZ lumping}\index{HRZ lumping}. 897 Lumping does not use the linear system solver library. 898 \end{memberdesc} 899 900 \begin{memberdesc}[SolverOptions]{PRES20} 901 the GMRES method with truncation after five residuals and restart after 902 20 steps, see \Ref{WEISS}. 903 \end{memberdesc} 904 905 \begin{memberdesc}[SolverOptions]{CGS} 906 conjugate gradient squared method, see \Ref{WEISS}. 907 \end{memberdesc} 908 909 \begin{memberdesc}[SolverOptions]{BICGSTAB} 910 stabilized bi-conjugate gradients methods, see \Ref{WEISS}. 911 \end{memberdesc} 912 913 \begin{memberdesc}[SolverOptions]{SSOR} 914 symmetric successive over-relaxation method, see \Ref{WEISS}. 915 Typically used as preconditioner but some linear solver libraries support 916 this as a solver. 917 \end{memberdesc} 918 919 \begin{memberdesc}[SolverOptions]{ILU0} 920 the incomplete LU factorization preconditioner with no fill-in, see \Ref{Saad}. 921 \end{memberdesc} 922 923 \begin{memberdesc}[SolverOptions]{JACOBI} 924 the Jacobi preconditioner, see \Ref{Saad}. 925 \end{memberdesc} 926 927 \begin{memberdesc}[SolverOptions]{AMG} 928 the algebraic multi grid method, see \Ref{AMG}. This method can be used as 929 linear solver method but is more robust when used as a preconditioner. 930 \end{memberdesc} 931 932 \begin{memberdesc}[SolverOptions]{GAUSS_SEIDEL} 933 the symmetric Gauss-Seidel preconditioner, see \Ref{Saad}. 934 \member{getNumSweeps()} is the number of sweeps used. 935 \end{memberdesc} 936 937 %\begin{memberdesc}[SolverOptions]{RILU} 938 %relaxed incomplete LU factorization preconditioner, see \Ref{RELAXILU}. 939 %This method is similar to the one used for \ILU but dropped elements are added 940 %to the main diagonal with the relaxation factor \member{getRelaxationFactor}. 941 %\end{memberdesc} 942 943 \begin{memberdesc}[SolverOptions]{REC_ILU} 944 recursive incomplete LU factorization preconditioner, see \Ref{RILU}. 945 This method is similar to the one used for \ILU but applies reordering during 946 the factorization. 947 \end{memberdesc} 948 949 \begin{memberdesc}[SolverOptions]{NO_REORDERING} 950 no reordering is used during factorization. 951 \end{memberdesc} 952 953 \begin{memberdesc}[SolverOptions]{DEFAULT_REORDERING} 954 the default reordering method during factorization. 955 \end{memberdesc} 956 957 \begin{memberdesc}[SolverOptions]{MINIMUM_FILL_IN} 958 applies reordering before factorization using a fill-in minimization strategy. 959 You have to check with the particular solver library or linear solver package 960 if this is supported. In any case, it is advisable to apply reordering on the 961 mesh to minimize fill-in. 962 \end{memberdesc} 963 964 \begin{memberdesc}[SolverOptions]{NESTED_DISSECTION} 965 applies reordering before factorization using a nested dissection strategy. 966 You have to check with the particular solver library or linear solver package 967 if this is supported. In any case, it is advisable to apply reordering on the 968 mesh to minimize fill-in. 969 \end{memberdesc} 970 971 \begin{memberdesc}[SolverOptions]{SUPER_LU} 972 the SuperLU library~\cite{SuperLU} is used as a solver. 973 \end{memberdesc} 974 975 \begin{memberdesc}[SolverOptions]{PASTIX} 976 the Pastix library~\cite{PASTIX} is used as a solver. 977 \end{memberdesc} 978 979 \begin{memberdesc}[SolverOptions]{NO_PRECONDITIONER} 980 no preconditioner is applied. 981 \end{memberdesc} 982 983 \begin{memberdesc}[SolverOptions]{DIRECT_INTERPOLATION} 984 direct interpolation in \AMG, see \cite{AMG} 985 \end{memberdesc} 986 \begin{memberdesc}[SolverOptions]{CLASSIC_INTERPOLATION} 987 classic interpolation in \AMG, see \cite{AMG} 988 \end{memberdesc} 989 \begin{memberdesc}[SolverOptions]{CLASSIC_INTERPOLATION_WITH_FF_COUPLING} 990 classic interpolation with enforced FF coupling in \AMG, see \cite{AMG} 991 \end{memberdesc} 992 993 \input{lumping}

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