[petsc-users] Dealing with 2nd Derivatives on Boundaries

[David Scott]

Hello,

I am solving Poisson's equation (actually Laplace's equation in this 
simple test case) on a 3D structured grid. The boundary condition in the 
first dimension is periodic. In the others there are Von Neumann 
conditions except for one surface where the second derivative is zero. I 
have specified DMDA_BOUNDARY_NONE in these two dimensions and deal with 
the boundary conditions by constructing an appropriate matrix. Here is 
an extract from the Fortran code:

               if (j==0) then
                 ! Von Neumann boundary conditions on y=0 boundary.
                 v(1) = 1
                   col(MatStencil_i, 1) = i
                   col(MatStencil_j, 1) = j
                   col(MatStencil_k, 1) = k
                 v(2) = -1
                   col(MatStencil_i, 2) = i
                   col(MatStencil_j, 2) = j+1
                   col(MatStencil_k, 2) = k
                 call MatSetValuesStencil(B, 1, row, 2, col, v, 
INSERT_VALUES, ierr)
               else if (j==maxl) then
                 ! Boundary condition on y=maxl boundary.
                 v(1) = 1
                   col(MatStencil_i, 1) = i
                   col(MatStencil_j, 1) = j
                   col(MatStencil_k, 1) = k
                 v(2) = -2
                   col(MatStencil_i, 2) = i
                   col(MatStencil_j, 2) = j-1
                   col(MatStencil_k, 2) = k
                 v(3) = 1
                   col(MatStencil_i, 3) = i
                   col(MatStencil_j, 3) = j-2
                   col(MatStencil_k, 3) = k
                 call MatSetValuesStencil(B, 1, row, 3, col, v, 
INSERT_VALUES, ierr)
               else if (k==0) then


Here the second clause deals with the second derivative on the boundary.

In order for this code to work I have to set the stencil width to 2 even 
though 'j-2' refers to an interior, non-halo
point in the grid. This leads to larger halo swaps than would be 
required if a stencil width of 1 could be used.

Is there a better way to encode the problem?

David

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