Optimal Meshes for Finite Elements of Arbitrary Order

TL;DR

This paper analyzes how to optimally triangulate a domain for finite element interpolation of arbitrary order, providing asymptotic error estimates and practical mesh design strategies for improved approximation accuracy.

Contribution

It introduces sharp asymptotic error estimates for optimal anisotropic triangulations with arbitrary polynomial degree finite elements, extending previous results and offering practical mesh design methods.

Findings

Optimal anisotropic triangulations minimize interpolation error asymptotically.

Error estimates involve invariant polynomials of derivatives of the function.

Practical strategies for mesh design achieve near-optimal interpolation error.

Abstract

Given a function f defined on a bidimensional bounded domain and a positive integer N, we study the properties of the triangulation that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the Lp norm and we consider Lagrange finite elements of arbitrary polynomial degree m-1. We establish sharp asymptotic error estimates as N tends to infinity when the optimal anisotropic triangulation is used, recovering the earlier results on piecewise linear interpolation, an improving the results on higher degree interpolation. These estimates involve invariant polynomials applied to the m-th order derivatives of f. In addition, our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. We partially extend our results to higher dimensions for finite elements on simplicial partitions of a domain of arbitrary dimension. Key words : anisotropic finite elements, adaptive meshes, interpolation, nonlinear approximation.