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1 % $Id$
2 \section{\LinearPDE Class}
3 \label{SEC LinearPDE}
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 $\Gamma$.
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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