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

revision 998 by ksteube, Wed Feb 14 04:40:49 2007 UTC revision 999 by gross, Tue Feb 27 08:12:37 2007 UTC
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11  \chapter{The module \linearPDEs}  \chapter{The module \linearPDEs}
12
\declaremodule{extension}{linearPDEs} \modulesynopsis{Linear partial pifferential equation handler}
The module \linearPDEs provides an interface to define and solve linear partial
differential equations within \escript. \linearPDEs does not provide any
solver capabilities in itself but hands the PDE over to
the PDE solver library defined through the \Domain of the PDE.
The general interface is provided through the \LinearPDE class. The
\AdvectivePDE which is derived from the \LinearPDE class
provides an interface to PDE dominated by its advective terms. The \Poisson
class which is also derived form the \LinearPDE class should be used
to define the Poisson equation \index{Poisson}.
13
14  \section{\LinearPDE Class}
15    \section{Linear Partial Differential Equations}
16  \label{SEC LinearPDE}  \label{SEC LinearPDE}
17
18  The \LinearPDE class is used to define a general linear, steady, second order PDE  The \LinearPDE class is used to define a general linear, steady, second order PDE
# Line 116  d^{contact}\hackscore{ik}=d^{contact}\ha Line 107  d^{contact}\hackscore{ik}=d^{contact}\ha
107  Note that different from the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$  Note that different from the scalar case~\eqn{LINEARPDE.SINGLE.4} now the coefficients $D$, $d$ abd $d^{contact}$
108  have to be inspected.  have to be inspected.
109
110  \section{\LinearPDE class}
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  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 selction of the most
126  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.
127
# Line 348  Lumping does not use the linear system s Line 353  Lumping does not use the linear system s
353  \begin{memberdesc}[LinearPDE]{PRES20}  \begin{memberdesc}[LinearPDE]{PRES20}
354  the GMRES method with truncation after five residuals and  the GMRES method with truncation after five residuals and
355  restart after 20 steps, see~\Ref{WEISS}.  restart after 20 steps, see~\Ref{WEISS}.
356  \end{memberdesc}[LinearPDE]{CR}  \end{memberdesc}
357
358  \begin{memberdesc}[LinearPDE]{CGS}  \begin{memberdesc}[LinearPDE]{CGS}
# Line 404  applies reordering before factorization Line 409  applies reordering before factorization
409  linear solver package if this is supported. In any case, it is advisable to apply reordering on the mesh to minimize fill-in.  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}  \end{memberdesc}
411
412  \section{The \Poisson Class}  \subsection{The \Poisson Class}
413  The \Poisson class provides an easy way to define and solve the Poisson  The \Poisson class provides an easy way to define and solve the Poisson
414  equation  equation
415  \begin{equation}\label{POISSON.1}  \begin{equation}\label{POISSON.1}
# Line 428  opens a Poisson equation on the \Domain Line 433  opens a Poisson equation on the \Domain
433  assigns new values to \var{f} and \var{q}.  assigns new values to \var{f} and \var{q}.
434  \end{methoddesc}  \end{methoddesc}
435
436  \section{The \Helmholtz Class}  \subsection{The \Helmholtz Class}
437  The \Helmholtz class defines the Helmholtz problem  The \Helmholtz class defines the Helmholtz problem
438  \begin{equation}\label{HZ.1}  \begin{equation}\label{HZ.1}
439  \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f  \omega \; u - (k\; u\hackscore{,j})\hackscore{,j} = f
# Line 452  opens a Helmholtz equation on the \Domai Line 457  opens a Helmholtz equation on the \Domai
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.  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}  \end{methoddesc}
459
460  \section{The \Lame Class}  \subsection{The \Lame Class}
461  The \Lame class defines a Lame equation problem:  The \Lame class defines a Lame equation problem:
462  \begin{equation}\label{LE.1}  \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}  -\mu (u\hackscore{i,j}+u\hackscore{j,i})+\lambda u\hackscore{k,k})\hackscore{j} = F\hackscore{i}-\sigma\hackscore{ij,j}

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