On low-dimensional approximation of function spaces of interior regularity

TL;DR

This paper introduces a novel boundary extension technique for constructing local approximation spaces in finite element methods, leveraging interior regularity to achieve exponential convergence with reduced dimensionality.

Contribution

It proposes a new method based on boundary approximation extensions, improving the efficiency of local space construction for Lipschitz domains.

Findings

Achieves exponential convergence in local approximation spaces.

Reduces the influence of spatial dimension on convergence rates.

Enables construction of local spaces via simpler variational problems.

Abstract

Many elliptic boundary value problems exhibit an interior regularity property, which can be exploited to construct local approximation spaces that converge exponentially within function spaces satisfying this property. These spaces can be used to define local ansatz spaces within the framework of generalised finite element methods, leading to a better relation between dimensionality and convergence order. In this paper, we present a new technique for the construction of such spaces for Lipschitz domains. Instead of the commonly used approach based on eigenvalue problems it relies on extensions of approximations performed on the boundary. Hence, it improves the influence of the spatial dimension on the exponential convergence and allows to construct the local spaces by solving the original kind of variational problems on easily structured domains.