Grading of Triangulations Generated by Bisection

TL;DR

This paper proves the existence of a regularized mesh function with grading two for triangulations generated by adaptive bisection, extending previous results to higher dimensions and ensuring stability of finite element projections.

Contribution

It generalizes the grading results of triangulations from 2D to arbitrary dimensions and establishes $H^1$-stability of $L^2$-projections for finite element spaces.

Findings

Existence of a regularized mesh function with grading two for adaptive bisection triangulations.

Extension of grading results from 2D to higher dimensions.

$H^1$-stability of $L^2$-projection for polynomial degrees less than seven.

Abstract

For triangulations generated by the adaptive bisection algorithm by Maubach and Traxler we prove existence of a regularized mesh function with grading two. This sharpens previous results in two dimensions for the newest vertex bisection and generalizes them to arbitrary dimensions. In combination with Diening et al. (2021) this yields $H^1$-stability of the $L^2$-projection onto Lagrange finite element spaces for all polynomial degrees and dimensions smaller than seven.