# Diff of /trunk/doc/user/linearPDE.tex

revision 1315 by gross, Tue Feb 27 08:12:37 2007 UTC revision 1316 by ksteube, Tue Sep 25 03:18:30 2007 UTC
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5  %               \url{http://www.access.edu.au  %
6  %         Primary Business: Queensland, Australia.  %           Copyright 2003-2007 by ACceSS MNRF
7  %   Licensed under the Open Software License version 3.0  %       Copyright 2007 by University of Queensland
9    %                http://esscc.uq.edu.au
10    %        Primary Business: Queensland, Australia
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16
17  \chapter{The module \linearPDEs}  \chapter{The module \linearPDEs}
18
# Line 88  In case of a single PDE and a single com Line 94  In case of a single PDE and a single com
94  \label{LINEARPDE.SINGLE.6}  \label{LINEARPDE.SINGLE.6}
95  n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u]  n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u]
96
97  In this case the the coefficient $d^{contact}$ and $y^{contact}$ are eaach \Scalar  In this case the the coefficient $d^{contact}$ and $y^{contact}$ are each \Scalar
98  both in the \FunctionOnContactZero or \FunctionOnContactOne.  both in the \FunctionOnContactZero or \FunctionOnContactOne.
99
100  The PDE is symmetrical \index{symmetrical} if  The PDE is symmetrical \index{symmetrical} if
# Line 110  have to be inspected. Line 116  have to be inspected.
116
117  \subsection{Classes}  \subsection{Classes}
118  \declaremodule{extension}{esys.escript.linearPDEs}  \declaremodule{extension}{esys.escript.linearPDEs}
119  \modulesynopsis{Linear partial pifferential equation handler}  \modulesynopsis{Linear partial differential equation handler}
120  The module \linearPDEs provides an interface to define and solve linear partial  The module \linearPDEs provides an interface to define and solve linear partial
121  differential equations within \escript. \linearPDEs does not provide any  differential equations within \escript. \linearPDEs does not provide any
122  solver capabilities in itself but hands the PDE over to  solver capabilities in itself but hands the PDE over to
# Line 122  class which is also derived form the \Li Line 128  class which is also derived form the \Li
128  to define the Poisson equation \index{Poisson}.  to define the Poisson equation \index{Poisson}.
129
130  \subsection{\LinearPDE class}  \subsection{\LinearPDE class}
131  This is the general class to define a linear PDE in \escript. We list a selction of the most  This is the general class to define a linear PDE in \escript. We list a selection of the most
132  important methods of the class only and refer to reference guide \ReferenceGuide for a complete list.  important methods of the class only and refer to reference guide \ReferenceGuide for a complete list.
133
134  \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0}  \begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0}
135  opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations}  opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations}
136  and \var{numSolutions} gives the number of equations and the number of solutiopn components.  and \var{numSolutions} gives the number of equations and the number of solution components.
137  If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations  If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations
138  and the number solutions, respctively, stay undefined until a coefficient is  and the number solutions, respectively, stay undefined until a coefficient is
139  defined.  defined.
140  \end{classdesc}  \end{classdesc}
141
# Line 140  defined. Line 146  defined.
146  \optional{, d}\optional{, y}  \optional{, d}\optional{, y}
147  \optional{, d_contact}\optional{, y_contact}  \optional{, d_contact}\optional{, y_contact}
148  \optional{, q}\optional{, r}}  \optional{, q}\optional{, r}}
149  assigns new values to coefficients. By dafault all values are assumed to be zero\footnote{  assigns new values to coefficients. By default all values are assumed to be zero\footnote{
150  In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The  In fact it is assumed they are not present by assigning the value \code{escript.Data()}. The
151  can by used by the solver library to reduce computational costs.  can by used by the solver library to reduce computational costs.
152  }  }
# Line 170  switches the debug mode to on. Line 176  switches the debug mode to on.
176  \end{methoddesc}  \end{methoddesc}
177
178  \begin{methoddesc}[LinearPDE]{isUsingLumping}{}  \begin{methoddesc}[LinearPDE]{isUsingLumping}{}
179  returns \True if \LUMPING is set as the solver for the system of lienar equations.  returns \True if \LUMPING is set as the solver for the system of linear equations.
180  Otherwise \False is returned.  Otherwise \False is returned.
181  \end{methoddesc}  \end{methoddesc}
182
# Line 189  returns the solver method and preconditi Line 195  returns the solver method and preconditi
195
196  \begin{methoddesc}[LinearPDE]{setSolverPackage}{\optional{package=LinearPDE.DEFAULT}}  \begin{methoddesc}[LinearPDE]{setSolverPackage}{\optional{package=LinearPDE.DEFAULT}}
197  Set the solver package to be used by PDE library to solve the linear systems of equations. The  Set the solver package to be used by PDE library to solve the linear systems of equations. The
198  specified package may not be supported by the PDE solver library. In this case, dependng on  specified package may not be supported by the PDE solver library. In this case, depending on
199  the PDE solver, the default solver is used or an exeption is thrown.  the PDE solver, the default solver is used or an exception is thrown.
200  If \var{package} is not specified, the default package of the PDE solver library is used.  If \var{package} is not specified, the default package of the PDE solver library is used.
201  \end{methoddesc}  \end{methoddesc}
202
# Line 202  returns the linear solver package curren Line 208  returns the linear solver package curren
208  \begin{methoddesc}[LinearPDE]{setTolerance}{\optional{tol=1.e-8}}:  \begin{methoddesc}[LinearPDE]{setTolerance}{\optional{tol=1.e-8}}:
209  resets the tolerance for solution. The actually meaning of tolerance is  resets the tolerance for solution. The actually meaning of tolerance is
210  depending on the underlying PDE library. In most cases, the tolerance  depending on the underlying PDE library. In most cases, the tolerance
211  will only consider the error from solving the discerete problem but will  will only consider the error from solving the discrete problem but will
212  not consider any discretization error.  not consider any discretization error.
213  \end{methoddesc}  \end{methoddesc}
214
# Line 321  the solver library of \finley, see \Sec{ Line 327  the solver library of \finley, see \Sec{
327
328  \begin{memberdesc}[LinearPDE]{ITERATIVE}  \begin{memberdesc}[LinearPDE]{ITERATIVE}
329  the default iterative method and preconditioner. The actually used method depends on the  the default iterative method and preconditioner. The actually used method depends on the
330  PDE solver library and the solver package been choosen. Typically, \PCG is used for symmetric PDEs  PDE solver library and the solver package been chosen. Typically, \PCG is used for symmetric PDEs
331  and \BiCGStab otherwise, both with \JACOBI preconditioner.  and \BiCGStab otherwise, both with \JACOBI preconditioner.
332  \end{memberdesc}  \end{memberdesc}
333
# Line 438  The \Helmholtz class defines the Helmhol Line 444  The \Helmholtz class defines the Helmhol
444  \label{HZ.1}  \label{HZ.1}
445  \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f  \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f
446
447   with natural boundary conditons   with natural boundary conditions
448  \label{HZ.2}  \label{HZ.2}
449  k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u  k\; u\hackscore{,j} n\hackscore{,j} = g- \alpha \; u
450
# Line 462  The \Lame class defines a Lame equation Line 468  The \Lame class defines a Lame equation
468  \label{LE.1}  \label{LE.1}
469  -\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j}  -\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j}
470
471  with natural boundary conditons:  with natural boundary conditions:
472  \label{LE.2}  \label{LE.2}
473  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}  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}
474

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