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% $Id$ |
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\chapter{The module \linearPDEsPack} |
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\declaremodule{extension}{linearPDEs} \modulesynopsis{Linear partial pifferential equation handler} |
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The module \linearPDEsPack provides an interface to define and solve linear partial |
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differential equations within \escript. \linearPDEsPack does not provide any |
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solver capabilities in itself but hand the PDE over to |
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the PDE solver library defined through the \Domain of the PDE. |
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The general interface is provided through the \LinearPDE class. The |
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\AdvectivePDE which is derived from the \LinearPDE class |
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provides an interface to PDE dominated by its advective terms. The \Poisson |
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class which is also derived form the \LinearPDE class should be used |
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to define the Poisson equation \index{Poisson}. |
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\section{\LinearPDE Class} |
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\label{SEC LinearPDE} |
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The \LinearPDE class is used to define a general linear, steady, second order PDE |
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for an unknown function $u$ on a given $\Omega$ defined through a \Domain object. |
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In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes |
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the outer normal field on $\Gamma$. |
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A single PDE with a solution with a single component the linear PDE is defined in the |
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following form: |
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\begin{equation}\label{LINEARPDE.SINGLE.1} |
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-(A\hackscore{jl} u\hackscore{,l}){,j}+(B\hackscore{j} u)\hackscore{,j}+C\hackscore{l} u\hackscore{,l}+D u =-X\hackscore{j,j}+Y \; . |
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\end{equation} |
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$u_{,j}$ 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. |
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The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the |
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\Function on the PDE or objects that can be converted into such \Data objects. |
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$A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar. |
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The following natural |
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boundary conditions are considered \index{boundary condition!natural} on $\Gamma$: |
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\begin{equation}\label{LINEARPDE.SINGLE.2} |
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n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;. |
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\end{equation} |
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Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are |
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each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the |
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solution at certain locations in the domain. They have the form |
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\begin{equation}\label{LINEARPDE.SINGLE.3} |
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u=r \mbox{ where } q>0 |
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\end{equation} |
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$r$ and $q$ are each \Scalar where $q$ is the characteristic function |
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\index{characteristic function} defining where the constraint is applied. |
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The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1} |
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or \eqn{LINEARPDE.SINGLE.2}. The PDE is symmetrical \index{symmetrical} if |
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\begin{equation}\label{LINEARPDE.SINGLE.4} |
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A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j} |
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\end{equation} |
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For a system of PDEs and a solution with several components the PDE has the form |
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\begin{equation}\label{LINEARPDE.SYSTEM.1} |
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-(A\hackscore{ijkl} u\hackscore{k,l}){,j}+(B\hackscore{ijk} u_k)\hackscore{,j}+C\hackscore{ikl} u\hackscore{k,l}+D\hackscore{ik} u_k =-X\hackscore{ij,j}+Y\hackscore{i} \; . |
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\end{equation} |
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$A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne. |
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The natural boundary conditions \index{boundary condition!natural} take the form: |
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\begin{equation}\label{LINEARPDE.SYSTEM.2} |
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n\hackscore{j}(A\hackscore{ijkl} u\hackscore{k,l}){,j}+(B\hackscore{ijk} u_k)+d\hackscore{ik} u_k=n\hackscore{j}-X\hackscore{ij}+y\hackscore{i} \;. |
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\end{equation} |
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The coefficient $d$ is a \RankTwo and $y$ is a |
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\RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form |
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\begin{equation}\label{LINEARPDE.SYSTEM.3} |
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u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 |
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\end{equation} |
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$r$ and $q$ are each \RankOne. Notice that at some locations not necessarily all components must |
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have a constraint. The system of PDEs is symmetrical \index{symmetrical} if |
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\begin{eqnarray}\label{LINEARPDE.SYSTEM.4} |
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A\hackscore{ijkl}=A\hackscore{klij} \\ |
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B\hackscore{ijk}=C\hackscore{kij} \\ |
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D\hackscore{ik}=D\hackscore{ki} \\ |
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d\hackscore{ik}=d\hackscore{ki} \ |
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\end{eqnarray} |
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\LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$ |
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in the domain $\Omega$. To specify the conditions across the discontinuity we are using the |
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generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution |
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defined as |
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\begin{equation}\label{LINEARPDE.SYSTEM.5} |
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J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij} |
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\end{equation} |
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For the case of single solution component and single PDE $J$ is defined |
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\begin{equation}\label{LINEARPDE.SINGLE.5} |
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J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j} |
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\end{equation} |
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In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the |
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discontinuity pointing from side 0 towards side 1. For a system of PDEs |
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the contact condition takes the form |
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\begin{equation}\label{LINEARPDE.SYSTEM.6} |
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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} \; . |
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\end{equation} |
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where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the |
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discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference |
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of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. |
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The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a |
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\RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne. |
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In case of a single PDE and a single component solution the contact condition takes the form |
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\begin{equation}\label{LINEARPDE.SINGLE.6} |
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n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u] |
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\end{equation} |
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In this case the the coefficient $d^{contact}$ and $y^{contact}$ are eaach \Scalar |
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both in the \FunctionOnContactZero or \FunctionOnContactOne. |
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\begin{classdesc}{LinearPDE}{domain,numEquations=0,numSolutions=0} |
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opens a linear, steady, second order PDE on the \Domain \var{domain}. \var{numEquations} |
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and \var{numSolutions} gives the number of equations and the number of solutiopn components. |
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If \var{numEquations} and \var{numSolutions} is non-positive, the number of equations |
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and the number solutions, respctively, stay undefined until a coefficient is |
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defined. |
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\end{classdesc} |
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\begin{methoddesc}[LinearPDE]{setValues}{arg1,arg2,...,argN} |
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assigns new values to coefficients \var{arg1}, \var{arg2}, $\ldots$ \var{argN}. |
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The coefficients must be one of the valid coefficient names |
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\var{A}, |
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\var{B}, |
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\var{C}, |
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\var{D}, |
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\var{X}, |
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\var{Y}, |
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\var{d}, |
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\var{y}, |
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\var{dcontact}, |
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\var{ycontact}, |
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\var{r}, |
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or \var{q}. |
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If the coefficient is not a \Data object, it is converted into a \Data object in the |
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appropriate \FunctionSpace. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{getCoefficient}{name} |
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return the value assigned to coefficient \var{name}. If \var{name} is not a valid name |
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an exception is raised. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{cleanCoefficients}{} |
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resets all coefficients to their initialization values. This method is useful to call when trying to save memory |
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as is releaces eqnerences to coefficients but keeping the differential operator. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{getShapeOfCoefficient}{name} |
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returns the shape of coefficient \var{name} even if no value has been assigned to it. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{getFunctionSpaceOfCoefficient}{name} |
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returns the \FunctionSpace of coefficient \var{name} even if no value has been assigned to it. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{hasCoefficient}{name} |
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returns \True if \var{name} is valid name of a coefficient |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{getFunctionSpaceForEquation}{} |
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returns \FunctionSpace of the right hand side which is identical to \FunctionSpace of |
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the result of the PDE operator. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{getFunctionSpaceForSolution}{} |
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returns \FunctionSpace of the solution of the PDE hich is identical to \FunctionSpace of |
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the result of argument of the PDE operator. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setDebugOn}{} |
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switches the debug mode to on. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setDebugOff}{} |
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switches the debug mode to on. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{debug}{} |
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returns \True if the debug mode is switched on. Otherwise it return \False. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setLumpingOn}{} |
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switches on lumping \index{lumping}. If lumping is switched on |
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the operator is |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setLumpingOff} |
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switches lumping off \index{lumping}. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setLumping}{flag=\False} |
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switches on lumping if \var{flag} is \True. Otherwise lumping is swiched off. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{isUsingLumping}{} |
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returns \True if lumping is switched on. Otherwise \False is returned. |
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\end{methoddesc} |
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\begin{memberdesc}[LinearPDE]{DEFAULT_METHOD} |
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default method to be used to solve the PDE. An appropriate method should be |
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chosen by the used PDE solver library. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{DIRECT} |
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direct linear solver~\Ref{SAAD} |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{CHOLEVSKY} |
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direct solver based on Cholevsky factorization (or similar), see~\Ref{SAAD}. The solver will require a symmetric PDE. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{PCG} |
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preconditioned conjugate gradient method, see~\Ref{WEISS}. The solver will require a symmetric PDE. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{GMRES} |
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the GMRES method, see~\Ref{WEISS}. Truncation and restart ar econtrolled by the parameters |
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\var{truncation} and \var{restart} of \method{getSolution}. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{PRES20} |
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the GMRES method with trunction after five residuals and |
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restart after 20 steps, see~\Ref{WEISS}. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{CR} |
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conjugate residual method, see~\Ref{WEISS}. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{CGS} |
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conjugate gradient squared method, see~\Ref{WEISS}. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{BICGSTAB} |
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stabilzed bi-conjugate gradients methods, see~\Ref{WEISS}. |
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\end{memberdesc} |
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\begin{memberdesc}[LinearPDE]{SSOR} |
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symmetric successive overrelaxtion method, see~\Ref{WEISS}. |
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\end{memberdesc} |
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\begin{methoddesc}[LinearPDE]{setSolverMethod}{solver=linearPDE.DEFAULT_METHOD} |
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sets the solver method to be used. It is pointed out that the PDE solver library |
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does not know the specified solver method but may choose a similar method. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setTolerance}{tol=1.e-8} |
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resets the tolerance for solution. The actually meaning of tolerance is |
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depending on the underlying PDE library. In most cases, the tolerance |
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will only consider the error from solving the discerete problem but will |
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not consider any discretization error. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{getTolerance}{} |
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returns the current tolerance of the solution |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{isSymmetric}{} |
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returns \True if the PDE has been indicated to be symmetric. |
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Otherwise \False is returned. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setSymmetryOn}{} |
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indicates that the PDE is symmetric. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setSymmetryOff}{} |
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indicates that the PDE is not symmetric. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setSymmetryTo}{flag=\False} |
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indicates that the PDE is symmetric if \var{flag}=\True |
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and indicates a non-symmetric PDE is \var{flag}=\False. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderOn}{} |
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switches on the reduction of polynomial order for the solution and |
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equation evaluation. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderOff}{} |
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switches off the reduction of polynomial order for the solution and |
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equation evaluation. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderTo}{flag=\False} |
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switches on the reduction of polynomial order for the solution and |
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equation evaluation if \var{flag}=\True. Otherwise |
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the order reduction is switched off. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderForSolutionOn}{} |
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switches on reduction of polynomial order for the solution. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderForSolutionOff}{} |
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switches off reduction of polynomial order for the solution. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderForSolutionTo}{flag=\False} |
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switches on the reduction of polynomial order for the solution |
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if \var{flag}=\True. Otherwise |
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the order reduction is switched off. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderForEquationOn}{} |
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switches on reduction of polynomial order for the equation evaluation. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderForEquationOff}{} |
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switches off reduction of polynomial order for the equation evaluation. |
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\end{methoddesc} |
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\begin{methoddesc}[LinearPDE]{setReducedOrderForEquationTo}{flag=\False} |
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switches on the reduction of polynomial order for the equation |
| 310 |
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evaluation if \var{flag}=\True. Otherwise |
| 311 |
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the order reduction is switched off. |
| 312 |
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\end{methoddesc} |
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| 314 |
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\begin{methoddesc}[LinearPDE]{getOperator}{} |
| 315 |
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returns the \Operator of the PDE. |
| 316 |
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\end{methoddesc} |
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| 318 |
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\begin{methoddesc}[LinearPDE]{getRightHandSide}{ignoreConstraint=\False} |
| 319 |
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returns the right hand side of the PDE as a \Data object. If |
| 320 |
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\var{ignoreConstraint}=\True the constraints are not considered |
| 321 |
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when building up the right hand side. |
| 322 |
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\end{methoddesc} |
| 323 |
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| 324 |
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\begin{methoddesc}[LinearPDE]{getSystem}{} |
| 325 |
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returns the \Operator and right hand side of the PDE. |
| 326 |
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\end{methoddesc} |
| 327 |
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| 328 |
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\begin{methoddesc}[LinearPDE]{getSolution}{option1,option2,...,optionN} |
| 329 |
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returns (an approximation of) the solution of the PDE. \var{options1}, \var{options2} |
| 330 |
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$\ldots$ \var{optionsN} are options handed over to the underlying PDE solver library. |
| 331 |
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\end{methoddesc} |
| 332 |
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| 333 |
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\begin{methoddesc}[LinearPDE]{getDomain}{} |
| 334 |
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returns the \Domain of the PDE. |
| 335 |
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\end{methoddesc} |
| 336 |
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| 337 |
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\begin{methoddesc}[LinearPDE]{getDim}{} |
| 338 |
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returns the spatial dimension of the PDE. |
| 339 |
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\end{methoddesc} |
| 340 |
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| 341 |
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\begin{methoddesc}[LinearPDE]{getNumEquations}{} |
| 342 |
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returns the number of equations. |
| 343 |
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\end{methoddesc} |
| 344 |
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| 345 |
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\begin{methoddesc}[LinearPDE]{getNumSolutions}{} |
| 346 |
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returns the number of components of the solution. |
| 347 |
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\end{methoddesc} |
| 348 |
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| 349 |
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\begin{methoddesc}[LinearPDE]{checkSymmetry}{verbose=\False} |
| 350 |
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returns \True if the PDE is symmetric and \False otherwise. |
| 351 |
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The method is very computational expensive and should only be |
| 352 |
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called for testing purposes. The symmetry flag is not altered. |
| 353 |
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If \var{verbose}=\True information about where symmetry is violated |
| 354 |
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are printed. |
| 355 |
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\end{methoddesc} |
| 356 |
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| 357 |
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\begin{methoddesc}[LinearPDE]{getFlux}{u} |
| 358 |
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returns the flux $J\hackscore{ij}$ \index{flux} for given solution \var{u} |
| 359 |
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defined by \eqn{LINEARPDE.SYSTEM.5} and \eqn{LINEARPDE.SINGLE.5}, respectively. |
| 360 |
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\end{methoddesc} |
| 361 |
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| 362 |
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\begin{methoddesc}[LinearPDE]{applyOperator}{u} |
| 363 |
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applies the PDE operator to \var{u} |
| 364 |
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\end{methoddesc} |
| 365 |
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| 366 |
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\begin{methoddesc}[LinearPDE]{getResidual}{u} |
| 367 |
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returns the residual when insering \var{u} into the PDE |
| 368 |
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\end{methoddesc} |
| 369 |
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|
| 370 |
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\section{\AdvectivePDE Class} |
| 371 |
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under construction |
| 372 |
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|
| 373 |
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\section{The \Poisson Class} |
| 374 |
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|
| 375 |
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The \Poisson class provides an easy way to define and solve the Poisson |
| 376 |
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equation |
| 377 |
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\begin{equation}\label{POISSON.1} |
| 378 |
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-u\hackscore{,ii}=f\; . |
| 379 |
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\end{equation} |
| 380 |
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with homogeneous boundary conditions |
| 381 |
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\begin{equation}\label{POISSON.2} |
| 382 |
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n\hackscore{i}u\hackscore{,i}=0 |
| 383 |
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\end{equation} |
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and homogeneous constraints |
| 385 |
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\begin{equation}\label{POISSON.3} |
| 386 |
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u=0 \mbox{ where } q>0 |
| 387 |
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\end{equation} |
| 388 |
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$f$ has to be a \Scalar in the \Function and $q$ must be |
| 389 |
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a \Scalar in the \SolutionFS. |
| 390 |
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|
| 391 |
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\begin{classdesc}{Poisson}{domain} |
| 392 |
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opens a Poisson equation on the \Domain domain. \Poisson is derived from \LinearPDE. |
| 393 |
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\end{classdesc} |
| 394 |
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\begin{methoddesc}[Poisson]{setValue}{f=escript.Data(),q=escript.Data()} |
| 395 |
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assigns new values to \var{f} and \var{q}. |
| 396 |
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\end{methoddesc} |
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