LinearPDE Class

The LinearPDE class is used to define a general linear, steady, second order PDE for an unknown function $u$ on a given $\Omega$ defined through a Domain object. In the following $\Gamma$ denotes the boundary of the domain $\Omega$. $n$ denotes the outer normal field on $\Gamma$.

For a single PDE with a solution with a single component the linear PDE is defined in the following form:

$$-(A\hackscore{jl} u\hackscore{,l})\hackscore{,j}-(B\hackscore{j} u)\hackscore{,j}+C\hackscore{l} u\hackscore{,l}+D u =-X\hackscore{j,j}+Y \; .$$ (58)

$u_{,j}$ 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. The coefficients $A$, $B$, $C$, $D$, $X$ and $Y$ have to be specified through Data objects in the general FunctionSpace on the PDE or objects that can be converted into such Data objects. $A$ is a rank-2 Data object, $B$, $C$ and $X$ are rank-1 Data object and $D$ and $Y$ are scalar. The following natural boundary conditions are considered on $\Gamma$:

$$n\hackscore{j}(A\hackscore{jl} u\hackscore{,l}+B\hackscore{j} u)+d u=n\hackscore{j}X\hackscore{j} + y \;.$$ (59)

Notice that the coefficients $A$, $B$ and $X$ are defined in the PDE. The coefficients $d$ and $y$ are each a scalar Data object in the boundary FunctionSpace. Constraints for the solution prescribing the value of the solution at certain locations in the domain. They have the form

$$u=r q>0$$ (60)

$r$ and $q$ are each scalar Data object where $q$ is the characteristic function defining where the constraint is applied. The constraints defined by Equation (4.3) override any other condition set by Equation (4.1) or Equation (4.2).

For a system of PDEs and a solution with several components the PDE has the form

$$-(A\hackscore{ijkl} u\hackscore{k,l})\hackscore{,j}-(B\hackscore{... ...ore{k,l}+D\hackscore{ik} u\hackscore{k} =-X\hackscore{ij,j}+Y\hackscore{i} \; .$$ (61)

$A$ is a rank-4 Data object, $B$ and $C$ are each a rank-3 Data object, $D$ and $X$ are each a rank-2 Data object and $Y$ is a rank-1 Data object. The natural boundary conditions take the form:

$$n\hackscore{j}(A\hackscore{ijkl} u\hackscore{k,l}+B\hackscore{ijk... ...d\hackscore{ik} u\hackscore{k}=n\hackscore{j}X\hackscore{ij}+y\hackscore{i} \;.$$ (62)

The coefficient $d$ is a rank-2 Data object and $y$ is a rank-1 Data object both in the boundary FunctionSpace. Constraints take the form

$$u\hackscore{i}=r\hackscore{i} q\hackscore{i}>0$$ (63)

$r$ and $q$ are each rank-1 Data object. Notice that not necessarily all components must have a constraint at all locations.

LinearPDE also supports solution discontinuities over contact region $\Gamma^{contact}$ in the domain $\Omega$. To specify the conditions across the discontinuity we are using the generalised flux $J$ which is in the case of a systems of PDEs and several components of the solution defined as

$$J\hackscore{ij}=A\hackscore{ijkl}u\hackscore{k,l}+B\hackscore{ijk}u\hackscore{k}-X\hackscore{ij}$$ (64)

For the case of single solution component and single PDE $J$ is defined

$$J\hackscore{j}=A\hackscore{jl}u\hackscore{,l}+B\hackscore{j}u\hackscore{k}-X\hackscore{j}$$ (65)

In the context of discontinuities $n$ denotes the normal on the discontinuity pointing from side 0 towards side 1. For a system of PDEs the contact condition takes the form

$$n\hackscore{j} J^{0}\hackscore{ij}=n\hackscore{j} J^{1}\hackscore{ij}=y^{contact}\hackscore{i} - d^{contact}\hackscore{ik} [u]\hackscore{k} \; .$$ (66)

where $J^{0}$ and $J^{1}$ are the fluxes on side 0 and side $1$ of the discontinuity $\Gamma^{contact}$, respectively. $[u]$, which is the difference of the solution at side 1 and at side 0, denotes the jump of $u$ across $\Gamma^{contact}$. The coefficient $d^{contact}$ is a rank-2 Data object and $y^{contact}$ is a rank-1 Data object both in the contact FunctionSpace on side 0 or contact FunctionSpace on side 1. In case of a single PDE and a single component solution the contact condition takes the form

$$n\hackscore{j} J^{0}\hackscore{j}=n\hackscore{j} J^{1}\hackscore{j}=y^{contact} - d^{contact}[u]$$ (67)

In this case the the coefficient $d^{contact}$ and $y^{contact}$ are eaach scalar Data object both in the contact FunctionSpace on side 0 or contact FunctionSpace on side 1.

The PDE is symmetrical if

$$A\hackscore{jl}=A\hackscore{lj} B\hackscore{j}=C\hackscore{j}$$ (68)

The system of PDEs is symmetrical if

$$A\hackscore{ijkl} =A\hackscore{klij}$$ (69)

$$B\hackscore{ijk}=C\hackscore{kij}$$ (70)

$$D\hackscore{ik}=D\hackscore{ki}$$ (71)

$$d\hackscore{ik}=d\hackscore{ki}$$ (72)

$$d^{contact}\hackscore{ik}=d^{contact}\hackscore{ki}$$ (73)

Note that different from the scalar case  Equation (4.11) now the coefficients $D$, $d$ abd $d^{contact}$ have to be inspected.