Convergence of Adaptive Mixed Finite Element Methods for the Hodge Laplacian Equation: Without Harmonic Forms

Abstract

Finite element exterior calculus (FEEC) has been developed as a systematic framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. This paper is devoted to analyzing the convergence of adaptive mixed finite element methods for the Hodge Laplacian equations based on FEEC without considering harmonic forms. A residual type a posteriori error estimate 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, based on which the convergence of adaptive mixed finite element methods is obtained without assuming the initial mesh size is small enough.