CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK

Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski

1. 1. Department of Mathematics, University of California, San Diego, CA 92123, USA;
2. Department of Mathematics and Statistics, Cal Poly Pomona, USA
• Supported by:
MS was partially supported by NSF Awards 1620366, 1262982, and 1217175. YL was partially supported by NSF Award 1620366. AM adn RS were partially supported by NSF Award 1217175.

Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski. CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK[J]. Journal of Computational Mathematics, 2020, 5(38): 748-767.

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. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poissonls equation on contractible domains in ${\Bbb R}$2, which can be viewed as a boundary problem on the de Rham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371) developed fundamental tools for a posteriori analysis on the de Rham complex. In this paper, we use tools in FEEC to construct convergence and complexity results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.

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