Meshes with symmetries

[JarosT55]

Hi,

Skfem offers predefined meshes with symmetries. Can & how this could be extended to custom meshes?

scipy.linalg.solve() offers assume_a parameter facilitating dealing with symmetric matrices.
Is there a way to trick or build in a skfem environment creating a similar mechanism for scipy.sparse.linalg.spsolve()?

Regards
MichaelT

[kinnala]

Symmetric mesh does not lead to a symmetric matrix. However, symmetric bilinear form (u and v can be replaced by one another) leads to a symmetric matrix.

As far as I know, spsolve is a wrapper to SuperLU which has no special support for symmetric matrices. The package scikit-sparse is a wrapper to CHOLMOD which uses the Cholesky decomposition instead of LU decomposition. Using Cholesky decomposition is faster for symmetric and positive definite matrices.

[JarosT55]

Hi Mr. Kinnala,

‘Symmetric mesh does not lead to a symmetric matrix.’

Yee, … I realized it by looking at how pre-built symmetrical tri-meshes are initialized.
Essentially, they are flat meshes containing ‘left’ and ‘right’ symmetrical halves.

Well, following such a concept and assembling a symmetrical custom mesh from its one decently sized half (e.g. using: $m_L = m_R.mirrored((0, 1, 0)); m_S = m_L + m_R$ ) would bring no benefit, but a memory and compute time blow-out.

The realistic, potentially beneficial approach (as I see it) would be ‘injecting’ symmetricity at the solver/boundary level … but I do not know how to do it … ⚡! The technicality/complexity of the method might be out of reach for Skfem Gringo, though.

Can Skfem offer other tricks from its symmetricity department?

Regards
MichaelT