# [petsc-users] Parallel dense solve with submatrices

Toon Weyens weyenst at gmail.com
Fri Aug 25 05:01:36 CDT 2017

```Dear all,

For a Bounday Element Method problem I require the solution of a system of
linear equations with multiple right-hand sides. Though this is a dense
system, I still want to do it via Petsc. Would the best way to do this be
through something such as

-ksp_type preonly -pc_type lu -pc_factor_mat_solver_package mumps  ?

Furthermore, I refine my grid in a regular way, which leads to a new system
of equations of size 2N x 2N where N is the original number of unknowns
before refining it. The upper-left matrix A_11 in this new system is
identical to the unrefined matrix (multiplied by 0.5). The other three
blocks A_12, A_21 and A_22 are new:

A_11 A_12
A_21 A_22

The question is now whether knowledge about the unrefined matrix A_11 can
be used to speed up calculation of the refined system?

I was thinking, for example, about using the LU decomposition of A_11 to
calculate the LU decomposition of the entire matrix A using the well-known
formula's

L_21 U_11 = A_21
L_11 U_12 = A_12
L_22 U_22 = A_22 - L_21 U_12

where L_21 and U_12 are full matrices, L_11, L_22 are lower triangular
matrices and U_11, U_22 are upper triangular.

Is there any way to do this in Petsc?

Or is there a better thing I can do?