The optimal aspect ratio for piecewise quadratic anisotropic finite element approximation

TL;DR

This paper determines the optimal aspect ratio for mesh adaptation in piecewise quadratic finite element approximation, extending existing knowledge from linear elements to quadratic elements for improved efficiency.

Contribution

It introduces the optimal aspect ratio criteria specifically for piecewise quadratic finite elements, filling a gap in mesh adaptation theory.

Findings

Provides explicit aspect ratio formulas for quadratic elements

Enhances mesh adaptation efficiency in finite element methods

Extends linear element results to quadratic case

Abstract

Mesh adaptation for finite element approximation is a procedure used in numerous applications. The use of thin and long anisotropic triangles improves the efficiency of the procedure. When piecewise linear finite elements are used, the aspect ratio for mesh adaptation is generally dictated by the absolute value of the (estimated) hessian matrix of the approximated function. We give in this paper the corresponding aspect ratio for piecewise quadratic finite elements.