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% $Id$ 
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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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For 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. 