Efficient preconditioning of hphp-FEM matrix sequences with slowly-varying coefficients: An application to topology optimization

We previously introduced a preconditioner that has proven effective for hphp-FEM discretizations of various challenging elliptic and hyperbolic problems. The construction is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi-Separable matrices.

The preconditioner is built as an approximate LDMtLDMt factorization through a divide-and-conquer approach. This implies an enhanced flexibility which allows to handle unstructured geometric meshes, anisotropies, and discontinuities. We build on our previous numerical experiments and develop a preconditioner-update strategy that allows us handle matrix sequences arising from problems with slowly-varying coefficients.

We investigate the performance of the preconditioner along with the update strategy in context of topology optimization of an acoustic cavity.