Convergence and quasi-optimality of adaptive finite element methods for harmonic forms

TL;DR

This paper proves that adaptive finite element methods reliably converge and achieve optimal rates when computing harmonic forms, regardless of the initial mesh quality, which is a significant advancement in numerical analysis for related fields.

Contribution

It establishes the convergence and quasi-optimality of AFEM for harmonic forms, even from coarse initial meshes, contrasting with prior results requiring fine initial meshes.

Findings

AFEM is contractive for harmonic forms

Achieves optimal convergence rates from any initial mesh

Contrasts with elliptic eigenvalue problem results

Abstract

Numerical computation of harmonic forms (typically called harmonic fields in three space dimensions) arises in various areas, including computer graphics and computational electromagnetics. The finite element exterior calculus framework also relies extensively on accurate computation of harmonic forms. In this work we study the convergence properties of adaptive finite element methods (AFEM) for computing harmonic forms. We show that a properly defined AFEM is contractive and achieves optimal convergence rate beginning from any initial conforming mesh. This result is contrasted with related AFEM convergence results for elliptic eigenvalue problems, where the initial mesh must be sufficiently fine in order for AFEM to achieve any provable convergence rate.