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 1 jgs 102 % $Id$ 2 gross 625 % 3 % Copyright © 2006 by ACcESS MNRF 4 % \url{http://www.access.edu.au 5 % Primary Business: Queensland, Australia. 6 % Licensed under the Open Software License version 3.0 7 8 % 9 jgs 102 10 gross 625 11 gross 599 \chapter{The module \linearPDEs} 12 jgs 102 13 14 gross 999 15 \section{Linear Partial Differential Equations} 16 jgs 102 \label{SEC LinearPDE} 17 18 The \LinearPDE class is used to define a general linear, steady, second order PDE 19 for an unknown function $u$ on a given $\Omega$ defined through a \Domain object. 20 In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes 21 gross 660 the outer normal field on $\Gamma$. 22 jgs 102 23 gross 660 For a single PDE with a solution with a single component the linear PDE is defined in the 24 jgs 102 following form: 25 \begin{equation}\label{LINEARPDE.SINGLE.1} 26 gross 660 -(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 \; . 27 jgs 102 \end{equation} 28 gross 660 $u_{,j}$ 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. 29 The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the 30 \Function on the PDE or objects that can be converted into such \Data objects. 31 $A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar. 32 jgs 102 The following natural 33 boundary conditions are considered \index{boundary condition!natural} on $\Gamma$: 34 \begin{equation}\label{LINEARPDE.SINGLE.2} 35 n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;. 36 \end{equation} 37 gross 660 Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are 38 each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the 39 jgs 102 solution at certain locations in the domain. They have the form 40 \begin{equation}\label{LINEARPDE.SINGLE.3} 41 u=r \mbox{ where } q>0 42 \end{equation} 43 $r$ and $q$ are each \Scalar where $q$ is the characteristic function 44 \index{characteristic function} defining where the constraint is applied. 45 The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1} 46 gross 660 or \eqn{LINEARPDE.SINGLE.2}. 47 gross 625 48 jgs 102 For a system of PDEs and a solution with several components the PDE has the form 49 \begin{equation}\label{LINEARPDE.SYSTEM.1} 50 gross 660 -(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} \; . 51 jgs 102 \end{equation} 52 gross 660 $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne. 53 jgs 102 The natural boundary conditions \index{boundary condition!natural} take the form: 54 \begin{equation}\label{LINEARPDE.SYSTEM.2} 55 gross 625 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} \;. 56 jgs 102 \end{equation} 57 gross 660 The coefficient $d$ is a \RankTwo and $y$ is a 58 jgs 102 \RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form 59 \begin{equation}\label{LINEARPDE.SYSTEM.3} 60 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 61 \end{equation} 62 gross 660 $r$ and $q$ are each \RankOne. Notice that not necessarily all components must 63 gross 625 have a constraint at all locations. 64 65 jgs 102 \LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$ 66 in the domain $\Omega$. To specify the conditions across the discontinuity we are using the 67 generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution 68 gross 660 defined as 69 jgs 102 \begin{equation}\label{LINEARPDE.SYSTEM.5} 70 J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij} 71 \end{equation} 72 For the case of single solution component and single PDE $J$ is defined 73 \begin{equation}\label{LINEARPDE.SINGLE.5} 74 J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j} 75 \end{equation} 76 gross 660 In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the 77 jgs 102 discontinuity pointing from side 0 towards side 1. For a system of PDEs 78 the contact condition takes the form 79 \begin{equation}\label{LINEARPDE.SYSTEM.6} 80 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} \; . 81 \end{equation} 82 where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the 83 discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference 84 of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. 85 gross 660 The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a 86 jgs 102 \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne. 87 In case of a single PDE and a single component solution the contact condition takes the form 88 \begin{equation}\label{LINEARPDE.SINGLE.6} 89 n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u] 90 \end{equation} 91 In this case the the coefficient $d^{contact}$ and $y^{contact}$ are eaach \Scalar 92 both in the \FunctionOnContactZero or \FunctionOnContactOne. 93 gross 625 94 The PDE is symmetrical \index{symmetrical} if 95 \begin{equation}\label{LINEARPDE.SINGLE.4} 96 A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j} 97 \end{equation} 98 The system of PDEs is symmetrical \index{symmetrical} if 99 \begin{eqnarray} 100 \label{LINEARPDE.SYSTEM.4} 101 A\hackscore{ijkl}=A\hackscore{klij} \\ 102 B\hackscore{ijk}=C\hackscore{kij} \\ 103 D\hackscore{ik}=D\hackscore{ki} \\ 104 d\hackscore{ik}=d\hackscore{ki} \\ 105 gross 660 d^{contact}\hackscore{ik}=d^{contact}\hackscore{ki} 106 gross 625 \end{eqnarray} 107 Note that different from the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$ 108 have to be inspected. 109 110 gross 999 111 \subsection{Classes} 112 \declaremodule{extension}{esys.escript.linearPDEs} 113 \modulesynopsis{Linear partial pifferential equation handler} 114 The module \linearPDEs provides an interface to define and solve linear partial 115 differential equations within \escript. \linearPDEs does not provide any 116 solver capabilities in itself but hands the PDE over to 117 the PDE solver library defined through the \Domain of the PDE. 118 The general interface is provided through the \LinearPDE class. The 119 \AdvectivePDE which is derived from the \LinearPDE class 120 provides an interface to PDE dominated by its advective terms. The \Poisson 121 class which is also derived form the \LinearPDE class should be used 122 to define the Poisson equation \index{Poisson}. 123 124 \subsection{\LinearPDE class} 125 gross 660 This is the general class to define a linear PDE in \escript. We list a selction of the most 126 gross 625 important methods of the class only and refer to reference guide \ReferenceGuide for a complete list. 127 128 jgs 102 \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0} 129 opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations} 130 and \var{numSolutions} gives the number of equations and the number of solutiopn components. 131 gross 660 If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations 132 jgs 102 and the number solutions, respctively, stay undefined until a coefficient is 133 gross 660 defined. 134 jgs 102 \end{classdesc} 135 136 gross 625 \begin{methoddesc}[LinearPDE]{setValue}{ 137 gross 660 \optional{A}\optional{, B}, 138 \optional{, C}\optional{, D} 139 \optional{, X}\optional{, Y} 140 \optional{, d}\optional{, y} 141 \optional{, d_contact}\optional{, y_contact} 142 \optional{, q}\optional{, r}} 143 assigns new values to coefficients. By dafault all values are assumed to be zero\footnote{ 144 In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The 145 can by used by the solver library to reduce computational costs. 146 } 147 gross 625 If the new coefficient value is not a \Data object, it is converted into a \Data object in the 148 jgs 102 appropriate \FunctionSpace. 149 \end{methoddesc} 150 151 \begin{methoddesc}[LinearPDE]{getCoefficient}{name} 152 gross 660 return the value assigned to coefficient \var{name}. If \var{name} is not a valid name 153 an exception is raised. 154 jgs 102 \end{methoddesc} 155 156 \begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name} 157 returns the shape of coefficient \var{name} even if no value has been assigned to it. 158 \end{methoddesc} 159 160 gross 625 \begin{methoddesc}[LinearPDE]{getFunctionSpaceForCoefficient}{name} 161 jgs 102 returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it. 162 \end{methoddesc} 163 164 \begin{methoddesc}[LinearPDE]{setDebugOn}{} 165 switches the debug mode to on. 166 \end{methoddesc} 167 168 \begin{methoddesc}[LinearPDE]{setDebugOff}{} 169 switches the debug mode to on. 170 \end{methoddesc} 171 172 gross 625 \begin{methoddesc}[LinearPDE]{isUsingLumping}{} 173 gross 653 returns \True if \LUMPING is set as the solver for the system of lienar equations. 174 Otherwise \False is returned. 175 jgs 102 \end{methoddesc} 176 177 gross 653 \begin{methoddesc}[LinearPDE]{setSolverMethod}{\optional{solver=LinearPDE.DEFAULT}\optional{, preconditioner=LinearPDE.DEFAULT}} 178 gross 625 sets the solver method and preconditioner to be used. It is pointed out that a PDE solver library 179 gross 660 may not know the specified solver method but may choose a similar method and preconditioner. 180 jgs 102 \end{methoddesc} 181 182 gross 653 \begin{methoddesc}[LinearPDE]{getSolverMethodName}{} 183 returns the name of the solver method and preconditioner which is currently been used. 184 \end{methoddesc} 185 186 \begin{methoddesc}[LinearPDE]{getSolverMethod}{} 187 returns the solver method and preconditioner which is currently been used. 188 \end{methoddesc} 189 190 \begin{methoddesc}[LinearPDE]{setSolverPackage}{\optional{package=LinearPDE.DEFAULT}} 191 gross 660 Set the solver package to be used by PDE library to solve the linear systems of equations. The 192 gross 653 specified package may not be supported by the PDE solver library. In this case, dependng on 193 the PDE solver, the default solver is used or an exeption is thrown. 194 gross 660 If \var{package} is not specified, the default package of the PDE solver library is used. 195 gross 653 \end{methoddesc} 196 197 \begin{methoddesc}[LinearPDE]{getSolverPackage}{} 198 returns the linear solver package currently by the PDE solver library 199 \end{methoddesc} 200 201 202 gross 625 \begin{methoddesc}[LinearPDE]{setTolerance}{\optional{tol=1.e-8}}: 203 resets the tolerance for solution. The actually meaning of tolerance is 204 gross 660 depending on the underlying PDE library. In most cases, the tolerance 205 gross 625 will only consider the error from solving the discerete problem but will 206 not consider any discretization error. 207 \end{methoddesc} 208 jgs 102 209 gross 625 \begin{methoddesc}[LinearPDE]{getTolerance}{} 210 returns the current tolerance of the solution 211 jgs 102 \end{methoddesc} 212 213 gross 625 \begin{methoddesc}[LinearPDE]{getDomain}{} 214 returns the \Domain of the PDE. 215 jgs 102 \end{methoddesc} 216 217 gross 625 \begin{methoddesc}[LinearPDE]{getDim}{} 218 returns the spatial dimension of the PDE. 219 jgs 102 \end{methoddesc} 220 221 gross 625 \begin{methoddesc}[LinearPDE]{getNumEquations}{} 222 returns the number of equations. 223 \end{methoddesc} 224 jgs 102 225 gross 625 \begin{methoddesc}[LinearPDE]{getNumSolutions}{} 226 returns the number of components of the solution. 227 jgs 102 \end{methoddesc} 228 229 gross 625 \begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False} 230 gross 660 returns \True if the PDE is symmetric and \False otherwise. 231 The method is very computational expensive and should only be 232 gross 625 called for testing purposes. The symmetry flag is not altered. 233 If \var{verbose}=\True information about where symmetry is violated 234 are printed. 235 jgs 102 \end{methoddesc} 236 237 gross 625 \begin{methoddesc}[LinearPDE]{getFlux}{u} 238 returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u} 239 defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively. 240 jgs 102 \end{methoddesc} 241 242 gross 625 243 jgs 102 \begin{methoddesc}[LinearPDE]{isSymmetric}{} 244 returns \True if the PDE has been indicated to be symmetric. 245 Otherwise \False is returned. 246 \end{methoddesc} 247 248 \begin{methoddesc}[LinearPDE]{setSymmetryOn}{} 249 indicates that the PDE is symmetric. 250 \end{methoddesc} 251 252 \begin{methoddesc}[LinearPDE]{setSymmetryOff}{} 253 indicates that the PDE is not symmetric. 254 \end{methoddesc} 255 256 \begin{methoddesc}[LinearPDE]{setReducedOrderOn}{} 257 gross 660 switches on the reduction of polynomial order for the solution and equation evaluation even if 258 a quadratic or higher interpolation order is defined in the \Domain. This feature may not 259 gross 625 be supported by all PDE libraries. 260 jgs 102 \end{methoddesc} 261 262 \begin{methoddesc}[LinearPDE]{setReducedOrderOff}{} 263 gross 660 switches off the reduction of polynomial order for the solution and 264 jgs 102 equation evaluation. 265 \end{methoddesc} 266 267 \begin{methoddesc}[LinearPDE]{getOperator}{} 268 returns the \Operator of the PDE. 269 \end{methoddesc} 270 271 gross 625 \begin{methoddesc}[LinearPDE]{getRightHandSide}{} 272 jgs 102 returns the right hand side of the PDE as a \Data object. If 273 \var{ignoreConstraint}=\True the constraints are not considered 274 when building up the right hand side. 275 \end{methoddesc} 276 277 \begin{methoddesc}[LinearPDE]{getSystem}{} 278 returns the \Operator and right hand side of the PDE. 279 \end{methoddesc} 280 281 gross 625 \begin{methoddesc}[LinearPDE]{getSolution}{ 282 \optional{verbose=False} 283 \optional{, reordering=LinearPDE.NO_REORDERING} 284 \optional{, iter_max=1000} 285 \optional{, drop_tolerance=0.01} 286 \optional{, drop_storage=1.20} 287 \optional{, truncation=-1} 288 \optional{, restart=-1} 289 } 290 gross 653 returns (an approximation of) the solution of the PDE. If \code{verbose=\True} some information during the solution process printed. 291 gross 660 \var{reordering} selects a reordering methods that is applied before or during the solution process 292 gross 653 (=\NOREORDERING ,\MINIMUMFILLIN ,\NESTEDDESCTION). 293 gross 660 \var{iter_max} specifies the maximum number of iteration steps that are allowed to reach the specified tolerance. 294 gross 625 \var{drop_tolerance} specifies a relative tolerance for small elements to be dropped when building a preconditioner 295 gross 653 (eg. in \ILUT). \var{drop_storage} limits the extra storage allowed when building a preconditioner 296 gross 660 (eg. in \ILUT). The extra storage is given relative to the size of the stiffness matrix, eg. 297 \var{drop_storage=1.2} will allow the preconditioner to use the $1.2$ fold storage space than used 298 for the stiffness matrix. \var{truncation} defines the truncation. 299 jgs 102 \end{methoddesc} 300 301 gross 625 \begin{memberdesc}[LinearPDE]{DEFAULT} 302 gross 660 default method, preconditioner or package to be used to solve the PDE. An appropriate method should be 303 gross 625 chosen by the used PDE solver library. 304 \end{memberdesc} 305 jgs 102 306 gross 625 \begin{memberdesc}[LinearPDE]{SCSL} 307 gross 660 the SCSL library by SGI,~\Ref{SCSL}\footnote{The SCSL library will only be available on SGI systems} 308 gross 625 \end{memberdesc} 309 jgs 102 310 gross 625 \begin{memberdesc}[LinearPDE]{MKL} 311 gross 653 the MKL library by Intel,~\Ref{MKL}\footnote{The MKL library will only be available when the intel compilation environment is used.}. 312 gross 625 \end{memberdesc} 313 jgs 102 314 gross 625 \begin{memberdesc}[LinearPDE]{UMFPACK} 315 gross 653 the UMFPACK,~\Ref{UMFPACK}. Remark: UMFPACK is not parallelized. 316 gross 625 \end{memberdesc} 317 jgs 102 318 gross 625 \begin{memberdesc}[LinearPDE]{PASO} 319 gross 653 the solver library of \finley, see \Sec{CHAPTER ON FINLEY}. 320 gross 625 \end{memberdesc} 321 jgs 102 322 gross 625 \begin{memberdesc}[LinearPDE]{ITERATIVE} 323 gross 653 the default iterative method and preconditioner. The actually used method depends on the 324 gross 660 PDE solver library and the solver package been choosen. Typically, \PCG is used for symmetric PDEs 325 and \BiCGStab otherwise, both with \JACOBI preconditioner. 326 gross 625 \end{memberdesc} 327 jgs 102 328 gross 625 \begin{memberdesc}[LinearPDE]{DIRECT} 329 gross 660 the default direct linear solver. 330 gross 625 \end{memberdesc} 331 jgs 102 332 gross 625 \begin{memberdesc}[LinearPDE]{CHOLEVSKY} 333 gross 660 direct solver based on Cholevsky factorization (or similar), see~\Ref{Saad}. The solver will require a symmetric PDE. 334 gross 625 \end{memberdesc} 335 jgs 110 336 gross 625 \begin{memberdesc}[LinearPDE]{PCG} 337 gross 653 preconditioned conjugate gradient method, see~\Ref{WEISS}\index{linear solver!PCG}\index{PCG}. The solver will require a symmetric PDE. 338 gross 625 \end{memberdesc} 339 jgs 110 340 gross 625 \begin{memberdesc}[LinearPDE]{GMRES} 341 gross 653 the GMRES method, see~\Ref{WEISS}\index{linear solver!GMRES}\index{GMRES}. Truncation and restart are controlled by the parameters 342 gross 625 \var{truncation} and \var{restart} of \method{getSolution}. 343 \end{memberdesc} 344 jgs 102 345 gross 625 \begin{memberdesc}[LinearPDE]{LUMPING} 346 gross 660 uses lumping to solve the system of linear equations~\index{linear solver!lumping}\index{lumping}. This solver technique 347 condenses the stiffness matrix to a diagonal matrix so the solution of the linear systems becomes very cheap. It can be used when 348 gross 653 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 349 gross 660 but is expect to converge to zero when the mesh gets finer. 350 Lumping does not use the linear system solver library. 351 gross 625 \end{memberdesc} 352 jgs 107 353 gross 625 \begin{memberdesc}[LinearPDE]{PRES20} 354 gross 653 the GMRES method with truncation after five residuals and 355 gross 625 restart after 20 steps, see~\Ref{WEISS}. 356 gross 999 \end{memberdesc} 357 gross 625 358 \begin{memberdesc}[LinearPDE]{CGS} 359 conjugate gradient squared method, see~\Ref{WEISS}. 360 jgs 107 \end{memberdesc} 361 362 gross 625 \begin{memberdesc}[LinearPDE]{BICGSTAB} 363 gross 660 stabilized bi-conjugate gradients methods, see~\Ref{WEISS}. 364 jgs 107 \end{memberdesc} 365 366 gross 625 \begin{memberdesc}[LinearPDE]{SSOR} 367 gross 653 symmetric successive over-relaxation method, see~\Ref{WEISS}. Typically used as preconditioner but some linear solver libraries support 368 gross 660 this as a solver. 369 gross 625 \end{memberdesc} 370 \begin{memberdesc}[LinearPDE]{ILU0} 371 gross 660 the incomplete LU factorization preconditioner with no fill-in, see~\Ref{Saad}. 372 gross 653 \end{memberdesc} 373 374 gross 625 \begin{memberdesc}[LinearPDE]{ILUT} 375 gross 653 the incomplete LU factorization preconditioner with fill-in, see~\Ref{Saad}. During the LU-factorization element with 376 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the 377 gross 660 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the 378 \method{getSolution} call. 379 gross 653 \end{memberdesc} 380 381 gross 625 \begin{memberdesc}[LinearPDE]{JACOBI} 382 gross 653 the Jacobi preconditioner, see~\Ref{Saad}. 383 \end{memberdesc} 384 385 gross 625 \begin{memberdesc}[LinearPDE]{AMG} 386 gross 660 the algebraic--multi grid method, see~\Ref{AMG}. This method can be used as linear solver method but is more robust when used 387 gross 653 in a preconditioner. 388 \end{memberdesc} 389 390 gross 625 \begin{memberdesc}[LinearPDE]{RILU} 391 gross 653 recursive incomplete LU factorization preconditioner, see~\Ref{RILU}. This method is similar to \ILUT but uses smoothing 392 between levels. During the LU-factorization element with 393 relative size less then \var{drop_tolerance} are dropped. Moreover, the size of the LU-factorization is restricted to the 394 gross 660 \var{drop_storage}-fold of the stiffness matrix. \var{drop_tolerance} and \var{drop_storage} are both set in the 395 \method{getSolution} call. 396 gross 653 \end{memberdesc} 397 jgs 107 398 gross 653 \begin{memberdesc}[LinearPDE]{NO_REORDERING} 399 no ordering is used during factorization. 400 \end{memberdesc} 401 gross 625 402 gross 653 \begin{memberdesc}[LinearPDE]{MINIMUM_FILL_IN} 403 applies reordering before factorization using a fill-in minimization strategy. You have to check with the particular solver library or 404 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 405 \end{memberdesc} 406 gross 625 407 \begin{memberdesc}[LinearPDE]{NESTED_DISSECTION} 408 gross 653 applies reordering before factorization using a nested dissection strategy. You have to check with the particular solver library or 409 linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in. 410 \end{memberdesc} 411 gross 625 412 gross 999 \subsection{The \Poisson Class} 413 jgs 102 The \Poisson class provides an easy way to define and solve the Poisson 414 equation 415 \begin{equation}\label{POISSON.1} 416 -u\hackscore{,ii}=f\; . 417 \end{equation} 418 with homogeneous boundary conditions 419 \begin{equation}\label{POISSON.2} 420 n\hackscore{i}u\hackscore{,i}=0 421 \end{equation} 422 and homogeneous constraints 423 \begin{equation}\label{POISSON.3} 424 u=0 \mbox{ where } q>0 425 \end{equation} 426 $f$ has to be a \Scalar in the \Function and $q$ must be 427 gross 660 a \Scalar in the \SolutionFS. 428 jgs 102 429 \begin{classdesc}{Poisson}{domain} 430 opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE. 431 \end{classdesc} 432 \begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()} 433 assigns new values to \var{f} and \var{q}. 434 \end{methoddesc} 435 gross 625 436 gross 999 \subsection{The \Helmholtz Class} 437 gross 660 The \Helmholtz class defines the Helmholtz problem 438 \begin{equation}\label{HZ.1} 439 \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f 440 \end{equation} 441 with natural boundary conditons 442 \begin{equation}\label{HZ.2} 443 k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u 444 \end{equation} 445 and constraints: 446 \begin{equation}\label{HZ.3} 447 u=r \mbox{ where } q>0 448 \end{equation} 449 $\omega$, $k$, $f$ have to be a \Scalar in the \Function, 450 $g$ and $\alpha$ must be a \Scalar in the \FunctionOnBoundary, 451 and $q$ and $r$ must be a \Scalar in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 452 gross 625 453 gross 660 \begin{classdesc}{Helmholtz}{domain} 454 opens a Helmholtz equation on the \Domain domain. \Helmholtz is derived from \LinearPDE. 455 \end{classdesc} 456 \begin{methoddesc}[Helmholtz]{setValue}{ \optional{omega} \optional{, k} \optional{, f} \optional{, alpha} \optional{, g} \optional{, r} \optional{, q}} 457 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. 458 \end{methoddesc} 459 460 gross 999 \subsection{The \Lame Class} 461 gross 660 The \Lame class defines a Lame equation problem: 462 \begin{equation}\label{LE.1} 463 -\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j} 464 \end{equation} 465 with natural boundary conditons: 466 \begin{equation}\label{LE.2} 467 n\hackscore{j}(\mu \; (u\hackscore{i,j}+u\hackscore{j,i})+\lambda*u\hackscore{k,k}) = f\hackscore{i}+n\hackscore{j}\sigma\hackscore{ij} 468 \end{equation} 469 and constraint 470 \begin{equation}\label{LE.3} 471 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 472 \end{equation} 473 $\mu$, $\lambda$ have to be a \Scalar in the \Function, 474 $F$ has to be a \Vector in the \Function, 475 $\sigma$ has to be a \Tensor in the \Function, 476 $f$ must be a \Vector in the \FunctionOnBoundary, 477 and $q$ and $r$ must be a \Vector in the \SolutionFS or must be mapped or interpolated into the particular \FunctionSpace. 478 gross 625 479 gross 660 \begin{classdesc}{Lame}{domain} 480 opens a Lame equation on the \Domain domain. \Lame is derived from \LinearPDE. 481 \end{classdesc} 482 \begin{methoddesc}[Lame]{setValue}{ \optional{lame_lambda} \optional{, lame_mu} \optional{, F} \optional{, sigma} \optional{, f} \optional{, r} \optional{, q}} 483 assigns new values to 484 \var{lame_lambda}, 485 \var{lame_mu}, 486 \var{F}, 487 \var{sigma}, 488 \var{f}, 489 \var{r} and 490 \var{q} 491 By default all values are set to be zero. 492 \end{methoddesc} 493

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