Hello,

Can somebody confirm this please:

The Laplace matrix of the pressure-Poisson equation is ALWAYS symmetric and positive definite (ignoring boundary conditions)!?!?

For a transient incompressible viscous fluid simulation I have the problem that the Laplace matrix get asymmetric after some time steps which implies that the ICCG solver breaks down.

I also found out that if the mesh has different element division in x and y, then the matrix gets also asymmetric.

If you had faced same or similar behaviours, I would be very thankful for your suggestions.

Boni

The short answer to your question is "No" because I can come up with one that is not, but it is difficult to say more without knowing the details of your discretization. If you are talking about the standard finite difference discretization, then nonuniform grid spacing could give a non-symmetric matrix (but the details are important). I suspect that similar statements hold for FEM, and I don't know about positive definiteness.

Jason, thanks for your help!

Here more details. I use the finite element method with Q1P0 elements. And faced that whenever the element division (not necessary the side length) of x-direction differs to the y-direction, then I get an asymmetric matrix.

How could be that possible since the construct AT*B*A should be symmetric. Isn't it? Or do I miss a mathematical rule?

If I want to use ICCG which assumes a symmetric matrix then I have to divide the mesh equaly. Not flexible!

Or can I force symmetry? I know this sounds very hard but I just want to know if it's possible.

Thank you for any comment!! Boni

After some math research I came to the conclusion that YES C^T*M_L^-1*C(Laplace matrix) must be symmetric.

C_T*C is for sure symmetric and if M_L^-1 is also symmetric then the Laplace matrix does.

Here the proof: Herefor M_L^-1 = M (inverse diagonal lumped mass matrix)

A = C^T * M * C A^T = (C^T * M *C)^T A^T = C * M^T * C^T A^T = C^T * M^T * C If M^T = M (M is symmetric) then A^T = C^T * M * C = A so A^T = A ... A is then also symmetric

The asymmetry in my case had a computational reason. Since M_L^-1 has huge values compared to C(C^T), my fortran real variables where not accurate enough. I moved to double precision.