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

• 作 者: Michael Holst Yuwen Li Adam Mihalik Ryan Szypowski
• 机 构: Department of Mathematics, University of California, San Diego, CA 92123, USA;Department of Mathemat
• 出 处: 计算数学（英文版）
• 发表日期: 2020年5期
• 关键词: Finite Element Exterior Calculus Adaptive finite element methods A posteriori er

Finite Element Exterior Calculus (FEEC) was developed by old,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 Poisson’s equation on contractible domains in R2,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.