$\mathcal{H}^2$-matrices for translation-invariant kernel functions

TL;DR

This paper introduces a modified $ abla$-matrix construction that exploits translation-invariance to reduce storage requirements in boundary element methods for elliptic PDEs, with broad applicability.

Contribution

It presents a novel $ abla$-matrix construction leveraging translation-invariance, significantly decreasing storage needs in boundary element methods.

Findings

Reduced storage complexity for $ abla$-matrices.

Applicable to boundary integral operators with translation-invariant kernels.

Simple assumptions suffice for complexity estimates.

Abstract

Boundary element methods for elliptic partial differential equations typically lead to boundary integral operators with translation-invariant kernel functions. Taking advantage of this property is fairly simple for particle methods, e.g., Nystrom-type discretizations, but more challenging if the supports of basis functions have to be taken into account. In this article, we present a modified construction for $\mathcal{H}^2$-matrices that uses translation-invariance to significantly reduce the storage requirements. Due to the uniformity of the boxes used for the construction, we need only a few uncomplicated assumptions to prove estimates for the resulting storage complexity.