High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates

TL;DR

This paper introduces a new adaptive finite element method for the Laplace-Beltrami operator on parametric surfaces, achieving optimal convergence rates based on surface and PDE regularity.

Contribution

It develops an AFEM with arbitrary polynomial degree on surfaces with Besov regularity, establishing convergence rates tied to surface and solution regularity.

Findings

Proves optimal convergence rates for the proposed AFEM.

Establishes a relation between surface regularity and PDE error decay.

Demonstrates the method's effectiveness on surfaces with Besov regularity.

Abstract

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally $W^1_\infty$ and piecewise in a suitable Besov class embedded in $C^{1,\alpha}$ with $\alpha \in (0,1]$. The idea is to have the surface sufficiently well resolved in $W^1_\infty$ relative to the current resolution of the PDE in $H^1$. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in $W^1_\infty$ and PDE error in $H^1$.