Published

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

Long Chen and Yongke Wu

SIAM J. Numer. Anal., 55(6), 2905–2929, 2017.

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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 any assumption on the
initial mesh.