H-matrix approximability of the inverses of FEM matrices

TL;DR

This paper demonstrates that the inverse of FEM stiffness matrices for elliptic problems can be efficiently approximated using H-matrices with exponential convergence, enabling accurate LU decompositions without mesh coupling.

Contribution

It provides a novel analysis showing exponential approximability of FEM inverse matrices by H-matrices, independent of mesh size and boundary conditions.

Findings

Exponential convergence in local block rank achieved.

H-matrix LU-decompositions are exponentially accurate.

Analysis covers mixed boundary conditions, unlike prior work.

Abstract

We study the question of approximability for the inverse of the FEM stiffness matrix for (scalar) second order elliptic boundary value problems by blockwise low rank matrices such as those given by the H-matrix format. We show that exponential convergence in the local block rank can be achieved. We also show that exponentially accurate LU-decompositions in the H-matrix format are possible for the stiffness matrices arising in the FEM. Unlike prior works, our analysis avoids any coupling of the block rank r and the mesh width h and also covers mixed Dirichlet-Neumann-Robin boundary conditions.