[petsc-users] Block unstructured grid

[Mathieu Dutour]

On Thu, 11 Mar 2021 at 13:52, Mark Adams <mfadams at lbl.gov> wrote:

> Mathieu,
> We have "FieldSplit" support for fields, but I don't know if it has ever
> been pushed to 1000's of fields so it might fall down. It might work.
> FieldSplit lets you manipulate the ordering, say field major (j) or node
> major (i).
>
I just looked at it and FieldSplit appears to be used in preconditioner so
not exactly relevant.

What was unsatisfactory?
> It sounds like you made a rectangular matrix A(1000,3e5) . Is that correct?
>
That is incorrect. The matrix is of size (N, N) with N = 1000 * 3e^5. It is
a square
matrix coming from an implicit scheme.

Since the other answer appears to have the same misunderstanding, let me try
to re-explain my point:
--- In many contexts we need a partial differential equation that is not
scalar.
For example, the shallow water equation has b = 3 fields: H, HU, HV. There
are other
examples like wave modelling where we have something like b = 1000 fields
(in a
discretization).
--- So, if we work with say an unstructured grid with N nodes then the
total number
of variables of the system will be N_tot = 3N or N_tot = 1000N.

The linear system has N_tot unknowns and N_tot equations. The entries
can be written as idx = (i , j) with 1 <= i <= b   and   1 <= j <= N.

Thus the non-zero entries in the matrix will be of two kinds:
--- (idx1, idx2)   with idx1 = (i , j) and idx2 = (i' , j) , 1 <= i, i' <=
b and 1 <= j <= N.
Together those define a block in the matrix.

--- (idx1, idx2) with idx1 = (i , j) and idx2 = (i, j'), 1<= i <= b and 1<=
j, j' <= N.
For each unknown idx1, there will be about 6 unknowns idx2 of this form.

Otherwise, the block matrices do not have the same coefficients, so a tensor
product approach does not appear to be workable.

  Mathieu

>
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