 # Contents of /trunk/doc/user/pdeinterface.tex

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 1 % $Id$ 2 \section{\LinearPDE Class} 3 \label{SEC LinearPDE} 4 5 The \LinearPDE class is used to define a general linear, steady, second order PDE 6 for an unknown function $u$ on a given $\Omega$ defined through a \Domain object. 7 In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes 8 the outer normal field on $\Gamma$. 9 10 For a single PDE with a solution with a single component the linear PDE is defined in the 11 following form: 12 \begin{equation}\label{LINEARPDE.SINGLE.1} 13 -(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 \; . 14 \end{equation} 15 $u_{,j}$ 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. 16 The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through \Data objects in the 17 \Function on the PDE or objects that can be converted into such \Data objects. 18 $A$ is a \RankTwo, $B$, $C$ and $X$ are \RankOne and $D$ and $Y$ are scalar. 19 The following natural 20 boundary conditions are considered \index{boundary condition!natural} on $\Gamma$: 21 \begin{equation}\label{LINEARPDE.SINGLE.2} 22 n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;. 23 \end{equation} 24 Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are 25 each a \Scalar in the \FunctionOnBoundary. Constraints \index{constraint} for the solution prescribing the value of the 26 solution at certain locations in the domain. They have the form 27 \begin{equation}\label{LINEARPDE.SINGLE.3} 28 u=r \mbox{ where } q>0 29 \end{equation} 30 $r$ and $q$ are each \Scalar where $q$ is the characteristic function 31 \index{characteristic function} defining where the constraint is applied. 32 The constraints defined by \eqn{LINEARPDE.SINGLE.3} override any other condition set by \eqn{LINEARPDE.SINGLE.1} 33 or \eqn{LINEARPDE.SINGLE.2}. The PDE is symmetrical \index{symmetrical} if 34 \begin{equation}\label{LINEARPDE.SINGLE.4} 35 A\hackscore{jl}=A\hackscore{lj} \mbox{ and } B\hackscore{j}=C\hackscore{j} 36 \end{equation} 37 For a system of PDEs and a solution with several components the PDE has the form 38 \begin{equation}\label{LINEARPDE.SYSTEM.1} 39 -(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} \; . 40 \end{equation} 41 $A$ is a \RankFour, $B$ and $C$ are each a \RankThree, $D$ and $X$ are each a \RankTwo and $Y$ is a \RankOne. 42 The natural boundary conditions \index{boundary condition!natural} take the form: 43 \begin{equation}\label{LINEARPDE.SYSTEM.2} 44 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} \;. 45 \end{equation} 46 The coefficient $d$ is a \RankTwo and $y$ is a 47 \RankOne both in the \FunctionOnBoundary. Constraints \index{constraint} take the form 48 \begin{equation}\label{LINEARPDE.SYSTEM.3} 49 u\hackscore{i}=r\hackscore{i} \mbox{ where } q\hackscore{i}>0 50 \end{equation} 51 $r$ and $q$ are each \RankOne. Notice that at some locations not necessarily all components must 52 have a constraint. The system of PDEs is symmetrical \index{symmetrical} if 53 \begin{eqnarray}\label{LINEARPDE.SYSTEM.4} 54 A\hackscore{ijkl}=A\hackscore{klij} \\ 55 B\hackscore{ijk}=C\hackscore{kij} \\ 56 D\hackscore{ik}=D\hackscore{ki} \\ 57 d\hackscore{ik}=d\hackscore{ki} \ 58 \end{eqnarray} 59 \LinearPDE also supports solution discontinuities \index{discontinuity} over contact region $\Gamma^{contact}$ 60 in the domain $\Omega$. To specify the conditions across the discontinuity we are using the 61 generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution 62 defined as 63 \begin{equation}\label{LINEARPDE.SYSTEM.5} 64 J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij} 65 \end{equation} 66 For the case of single solution component and single PDE $J$ is defined 67 \begin{equation}\label{LINEARPDE.SINGLE.5} 68 J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j} 69 \end{equation} 70 In the context of discontinuities \index{discontinuity} $n$ denotes the normal on the 71 discontinuity pointing from side 0 towards side 1. For a system of PDEs 72 the contact condition takes the form 73 \begin{equation}\label{LINEARPDE.SYSTEM.6} 74 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} \; . 75 \end{equation} 76 where $J^{0}$ and $J^{1}$ are the fluxes on side $0$ and side $1$ of the 77 discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference 78 of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. 79 The coefficient $d^{contact}$ is a \RankTwo and $y^{contact}$ is a 80 \RankOne both in the \FunctionOnContactZero or \FunctionOnContactOne. 81 In case of a single PDE and a single component solution the contact condition takes the form 82 \begin{equation}\label{LINEARPDE.SINGLE.6} 83 n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u] 84 \end{equation} 85 In this case the the coefficient $d^{contact}$ and $y^{contact}$ are eaach \Scalar 86 both in the \FunctionOnContactZero or \FunctionOnContactOne.

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