Convergence of Adaptive Mixed Finite Element Methods for the Hodge Laplacian Equation: without harmonic forms

TL;DR

This paper proves the convergence of adaptive mixed finite element methods for the Hodge Laplacian equation using finite element exterior calculus, without the need for harmonic forms or small initial meshes.

Contribution

It introduces a residual a posteriori error estimate and a marking strategy that guarantees uniform convergence without small initial mesh assumptions.

Findings

Established residual a posteriori error estimates using Hodge decomposition.

Proved uniform convergence of adaptive methods without small initial mesh constraints.

Developed a marking strategy ensuring quasi-orthogonality.

Abstract

Finite element exterior calculus (FEEC) has been developed as a systematical framework for constructing and analyzing stable and accurate numerical method for partial differential equations by employing differential complexes. This paper is devoted to analyze convergence of adaptive mixed finite element methods for Hodge Laplacian equations based on FEEC without considering harmonic forms. More precisely, a residual type posteriori error estimates is obtained by using the Hodge decomposition, the regular decomposition and bounded commuting quasi-interpolants. An additional marking strategy is added to ensure the quasi-orthogonality. Using this quasi-orthogonality, the uniform convergence of adaptive mixed finite element methods is obtained without assuming the initial mesh size is small enough.