Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra

TL;DR

This paper establishes weighted anisotropic analytic regularity estimates for solutions of second order elliptic boundary value problems in polygons and polyhedra, enhancing understanding of solution smoothness near corners and edges.

Contribution

It provides a simplified proof of weighted analytic regularity in polygons and extends it to polyhedra, incorporating anisotropic estimates that account for edge regularity.

Findings

Weighted anisotropic analytic estimates are proven for polyhedral domains.

The method simplifies existing proofs and extends regularity results to three dimensions.

The approach does not require analyzing singular functions, simplifying the analysis.

Abstract

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.