Average Interpolation Under the Maximum Angle Condition

TL;DR

This paper develops new interpolation error estimates for finite element methods that work with less smooth functions and more general mesh shapes, including the maximum angle condition, by localizing error analysis.

Contribution

It introduces a localized approach to error estimation that allows simultaneous relaxation of smoothness and shape restrictions on simplicial meshes.

Findings

Error estimates valid for minimal smoothness functions.

Mesh shape restrictions can be relaxed under the maximum angle condition.

Estimates applicable to L^p and W^{1,p} data with minimal restrictions.

Abstract

Interpolation error estimates needed in common finite element applications using simplicial meshes typically impose restrictions on the both the smoothness of the interpolated functions and the shape of the simplices. While the simplest theory can be generalized to admit less smooth functions (e.g., functions in H^1(\Omega) rather than H^2(\Omega)) and more general shapes (e.g., the maximum angle condition rather than the minimum angle condition), existing theory does not allow these extensions to be performed simultaneously. By localizing over a well-shaped auxiliary spatial partition, error estimates are established under minimal function smoothness and mesh regularity. This construction is especially important in two cases: L^p(\Omega) estimates for data in W^{1,p}(\Omega) hold for meshes without any restrictions on simplex shape, and W^{1,p}(\Omega) estimates for data in W^{2,p}(\Omega) hold under a generalization of the maximum angle condition which previously required p>2 for standard Lagrange interpolation.