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CONVERGENCE AND OPTIMALITY OF ADAPTIVE METHODS IN THE FINITE ELEMENT EXTERIOR CALCULUS FRAMEWORK

By Michael Holst, Adam Mihalik and Ryan Szypowski

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 erro

Topics: CONTENTS
Year: 1306
OAI identifier: oai:CiteSeerX.psu:10.1.1.310.8765
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