[Paper Review] Convergence and Optimality of Adaptive Methods in the Finite Element Exterior Calculus Framework

Michael Holst, Adam Mihalik+1 more|arXiv (Cornell University)|Jun 8, 2013

Advanced Numerical Methods in Computational Mathematics26 references3 citations

TL;DR

This paper establishes convergence and optimality of adaptive finite element methods within the Finite Element Exterior Calculus (FEEC) framework by leveraging a quasi-orthogonality property derived from the Hodge decomposition. It extends prior results by proving that adaptive algorithms driven by a posteriori error indicators converge and achieve optimal convergence rates for mixed methods on Hilbert complexes.

ABSTRACT

ABSTRACT. Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert Complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Stability and consistency of the resulting methods then follow directly from the framework by establishing the existence of operators connecting the Hilbert complex with its subcomplex, giving a essentially a “recipe ” for well-behaved methods. In 2012, Demlow and Hirani developed a posteriori error indicators for driving adaptive methods in the FEEC framework. While adaptive techniques have been used successfully with mixed methods for years, convergence theory for such techniques has not been fully developed. The main difficulty is lack of error orthogonality. In 2009, Chen, Holst, and Xu established convergence and optimality of an adaptive mixed finite element method for the Poisson equation (the Hodge-Laplace problem fork = n = 2) on simply connected polygonal domains in two dimensions. Their argument used a type of quasi-orthogonality result, exploiting the fact that the error was orthogonal to the divergence free subspace, while the part of the error

Motivation & Objective

• To close the gap in convergence theory for adaptive mixed finite element methods, which lack error orthogonality.
• To extend the convergence and optimality results of Chen, Holst, and Xu (2009) from the Poisson equation to the broader FEEC framework.
• To establish that adaptive methods based on a posteriori error indicators in FEEC converge and achieve optimal convergence rates.
• To demonstrate that the quasi-orthogonality property, arising from the Hodge decomposition, enables convergence despite the absence of standard error orthogonality.
• To provide a theoretical foundation for adaptive mixed methods in complex geometries and general mixed variational problems within the FEEC framework.

Proposed method

• Utilizes the FEEC framework to model mixed variational problems on Hilbert complexes and their finite-dimensional subcomplexes.
• Employs a posteriori error indicators developed by Demlow and Hirani (2012) to drive adaptive mesh refinement.
• Establishes a quasi-orthogonality result by exploiting the Hodge decomposition, where the error is orthogonal to the divergence-free subspace.
• Applies the discrete inf-sup condition and stability of the FEEC framework to ensure well-posedness of the discrete problems.
• Uses the existence of projection operators connecting the Hilbert complex and its subcomplex to derive consistency and stability.
• Combines the quasi-orthogonality with a priori and a posteriori error estimates to prove convergence and optimality of the adaptive algorithm.

Experimental results

Research questions

• RQ1 Can adaptive mixed finite element methods in the FEEC framework achieve convergence despite the absence of standard error orthogonality?
• RQ2 What conditions ensure optimal convergence rates for adaptive FEM within the FEEC framework?
• RQ3 How can the quasi-orthogonality property be leveraged to prove convergence in the absence of full error orthogonality?
• RQ4 To what extent do a posteriori error indicators in FEEC lead to optimal adaptive refinement strategies?
• RQ5 Can the convergence and optimality theory from the Poisson equation be generalized to broader classes of mixed problems in FEEC?

Key findings

• The adaptive finite element method based on FEEC and a posteriori error indicators converges globally in the energy norm.
• Optimal convergence rates are achieved, meaning the error decays at the best possible rate for the given mesh refinement strategy.
• The quasi-orthogonality property, derived from the Hodge decomposition, enables convergence by ensuring that the error decreases over successive adaptive steps.
• The convergence and optimality results hold for general mixed variational problems formulated on Hilbert complexes, not just the Poisson equation.
• The framework ensures stability and consistency of the discrete methods through the existence of projection operators linking the continuous and discrete complexes.
• The theoretical results generalize prior work by Chen, Holst, and Xu (2009), extending their convergence and optimality analysis beyond the two-dimensional Poisson case.