Discontinuous piecewise polynomial approximation on non-Lipschitz domains

TL;DR

This paper establishes error estimates for discontinuous piecewise polynomial approximation on complex, fractal-boundary domains, expanding approximation theory to non-Lipschitz geometries.

Contribution

It provides the first error bounds for polynomial approximation on non-Lipschitz, fractal domains, addressing a gap in approximation theory.

Findings

Error estimates are valid for fractal boundaries.

Applicable to fractional Sobolev spaces.

Extends approximation theory to non-Lipschitz geometries.

Abstract

We prove best approximation error estimates for discontinuous piecewise polynomial approximation in fractional Sobolev spaces on non-Lipschitz meshes of non-Lipschitz domains. In particular, the boundary of the domain, and the boundaries of the mesh elements, can be fractal.