pdebound (Partial Differential Equations Toolbox)

Partial Differential Equations Toolbox

Boundary M-file.

[q,g,h,r]=pdebound(p,e,u,time)

Description
The Boundary M-file specifies the boundary conditions of a PDE problem.

The most general form of boundary conditions that we can handle is

hu = r

· (c u) + u = g + h'µ.

By the notation ·(c u) we mean the N-by-1 matrix with (i,1)-component

where the outward normal vector of the boundary

. There are M Dirichlet conditions and the h-matrix is M-by-N, M

0. The generalized Neumann condition contains a source h'm where the Lagrange multipliers µ is computed such that the Dirichlet conditions become satisfied.

The data that you specify is q, g, h, and r.

For M = 0 we say that we have a generalized Neumann boundary condition, for M = N a Dirichlet boundary condition, and for 0 < M < N a mixed boundary condition.

The Boundary M-file [q,g,h,r]=pdebound(p,e,u,time) computes the values of q, g, h, and r, on the a set of edges e.

The matrices p and e are mesh data. e needs only to be a subset of the edges in the mesh. Details on the mesh data representation can be found in the entry on initmesh.

The input arguments u and time are used for the nonlinear solver and time stepping algorithms, respectively. u and time are empty matrices if the corresponding parameter is not passed to assemb. If time is NaN and any of the function q, g, h, and r depends on time, pdebound must return a matrix of correct size, containing NaNs in all positions, in the corresponding output argument.

The solution u is represented by the solution vector u. Details on the representation can be found in the entry on assempde.

q and g must contain the value of q and g on the midpoint of each boundary. Thus we have size(q)=[N^2 ne], where N is the dimension of the system, and ne the number of edges in e, and size(g)=[N ne]. For the Dirichlet case, the corresponding values must be zeros.

h and r must contain the values of h and r at the first point on each edge followed by the value at the second point on each edge. Thus we have size(h)=[N^2 2*ne], where N is the dimension of the system, and ne the number of edges in e, and size(r)=[N 2*ne]. When M < N, h and r must be padded with N - M rows of zeros.

The elements of the matrices q and h are stored in column-wise ordering in the MATLAB matrices q and h.

Examples
For the boundary conditions

the values below should be stored in q, g, h, and r:

        1 
q=[ ... 2 ... ]
        2
        0

g=[ ... 3 ... ]
        4

        1     1 
h=[ ... 0 ... 0 ... ] 
       -1    -1
        0     0

r=[ ... 2 ... 2 ... ]
        0     0

See Also
initmesh, pdegeom, pdesdt, pdeent

pdearcl pdecgrad