Exponential meshes and $\mathcal{H}$-matrices

TL;DR

This paper extends previous results on exponential approximation of FEM inverse matrices using $\\mathcal{H}$-matrices, covering more mesh types, including graded meshes, and clarifies the role of polynomial degree.

Contribution

It broadens the class of meshes for which exponential $\\mathcal{H}$-matrix approximation applies and makes the dependence on polynomial degree explicit.

Findings

Approximation error bounds are sharpened.

The analysis now includes exponentially graded meshes.

Dependence on polynomial degree $p$ is explicitly characterized.

Abstract

In our previous works, we proved that the inverse of the stiffness matrix of an $h$-version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block rank by $\mathcal{H}$-matrices. Here, we improve on this result in multiple ways: (1) The class of meshes is significantly enlarged and includes certain exponentially graded meshes. (2) The dependence on the polynomial degree $p$ of the discrete ansatz space is made explicit in our analysis. (3) The bound for the approximation error is sharpened, and (4) the proof is simplified.