[petsc-users] Coupled system of equations on unstructured mesh

[Karthikeyan Chockalingam - STFC UKRI]

I will solve the coupled problem in a non-linear fashion using SNES, so I will have one residual for each variable R(u1) and R(u2) and the resulting Jacobian matrix

J =  [J11 J12
         J21 J22] for each element.

For first-order Lagrange elements (in 2D) will have four nodes, then J would be of size 8 x 8. Let's say the variables are u and v. The order would be [u1 u2 u3 u4 v1 v2 v3 v4].

Or should I rearrange the above entrées in the element Jacobian so as to have [u1 v1 u2 v2 u3 v3 u4 v4] ordering? In that case, the Jacobian will not have block ordering I guess.

In my case, the discretization of the u and v are from the same element space.
Can you please elaborate when you say, “Just enlarge the block size of your matrix”?
I have never used block size before and not sure what it accomplishes.
Do you mean to say, I should use MATBAIJ instead of MATAIJ?

Kind regards,
Karthik.

From: Matthew Knepley <knepley at gmail.com>
Date: Sunday, 10 September 2023 at 19:03
To: Chockalingam, Karthikeyan (STFC,DL,HC) <karthikeyan.chockalingam at stfc.ac.uk>
Cc: petsc-users at mcs.anl.gov <petsc-users at mcs.anl.gov>
Subject: Re: [petsc-users] Coupled system of equations on unstructured mesh
On Sun, Sep 10, 2023 at 1:48 PM Karthikeyan Chockalingam - STFC UKRI via petsc-users <petsc-users at mcs.anl.gov<mailto:petsc-users at mcs.anl.gov>> wrote:
Hello,

I have so far solved scalar field problems using finite elements on a given (unstructured) mesh. I made use of MATMPIAIJ to create matrixes, MatCreateVecs(A, &b, &x) to create vectors, and MatZeroRowsColumnsIS to set boundary conditions.

Now, I would like to solve a coupled system of equations for the quantities u1 and u2 on the (unstructured)  mesh. I.e., the matrix should get the double number of rows and columns,

A = [A00  A01
        A10  A11]

This is usually not a good way to think of it. This division means that all variables of one field come before
all those of another. It is much more common to group together all the unknowns at a given point.

You could, if the discretizations of u1 and u2 are the same, just enlarge the blocksize of your matrix. Then you set u1 and u2 for each vec location, or a 2x2 block for each Jacobian location.

  Thanks,

     Matt

the vectors contain twice the number of entries (e.g. first all u1s and then all u2s). I would like to be sure that the entries of u1 and u2, which are associated with the same element are located on the same processor.

Is a pre-defined structure already available within PETSc to enlarge such a single equation to store the entries of coupled equations?

-\Delta u_1+c_{11} u_1+c_{12} u_2=f_1
-\Delta u_2+c_{21} u_1+c_{22} u_2=f_2

Would I still be able to use MatZeroRowsColumnsIS u1 and u2 independently to enforce boundary conditions? MatZeroRowsColumnsIS(A, is, 1, x, b);

I don’t know where to begin. I have so far only been exposed to using MATMPIAIJ and MatSetValues to create and assign values to matrix entries respectively. I would be grateful if you could provide the stepwise guide.

Kind regards,
Karthik.

--
Karthik Chockalingam, Ph.D.
Senior Research Software Engineer
High Performance Systems Engineering Group
Hartree Centre | Science and Technology Facilities Council
karthikeyan.chockalingam at stfc.ac.uk<mailto:karthikeyan.chockalingam at stfc.ac.uk>

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--
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https://www.cse.buffalo.edu/~knepley/<http://www.cse.buffalo.edu/~knepley/>
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