Some Convergence and Optimality Results of Adaptive Mixed Methods in Finite Element Exterior Calculus

Abstract

In this paper, we present several new a posteriori error estimators and two adaptive mixed finite element methods AMFEM1 and AMFEM2 for the Hodge Laplacian problem in finite element exterior calculus. We prove that AMFEM1 and AMFEM2 are both convergent starting from any initial coarse mesh. In addition, we prove the quasi-optimality of AMFEM2. Comparing to existing literature, our results work on Lipschitz domains with nontrivial cohomology and provide the first norm convergence and quasi-optimality results for the Hodge Laplacian. Key words. a posteriori error estimate, adaptive mixed finite element method, finite element exterior calculus, Hodge Laplacian, convergence, optimality AMS subject classifications. 65N12, 65N15, 65N30, 65N50, 41A25 ∇ − → H(curl, Ω) curl −−→ H(div, Ω) div − − → L 2 (Ω).

Superconvergent Recovery of Raviart–Thomas Mixed Finite Elements on Triangular Grids

“…Since Ω is simply connected, (3.7) is exact and the discrete Helmholtz/Hodge decomposition (see, e.g., [1,2,9,18]) holds:…”

An Extended Galerkin analysis in finite element exterior calculus

“…For a thorough introduction to FEEC, readers are referred to [7,9,2] and references therein. As a model problem in FEEC, the continuous Galerkin/Arnold-Falk-Winther (AFW) method for the following Hodge-Laplace equation (without harmonic forms) (1.1) (dδ + δd)u = f has been discussed in many aspects, see, e.g., [7,20,30,45] for commuting projections and a priori error estimates, [27,26,18,43,44] for a posteriori error estimates and adaptive algorithms, [32,6] for time-dependent problems, and [3,33,28] for FEEC on cubical and polyhedral meshes. The discontinuous Galerkin (DG) methods can be traced back to the late 1960s [46,10].…”

Quasi-optimal adaptive hybridized mixed finite element methods for linear elasticity

“…For second order elliptic equations in mixed form, theoretical analysis of AMFEMs is extensive, see, e.g., [6,12,18,23,27,[31][32][33]. The optimality result of adaptive mixed methods for elasticity equations seems limited in the literature.…”