2013/06/08 by Michael Holst, Yuwen Li, Holst, Michael +5 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1306.1886
openalex publication_date 2013/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk,\nWinther and others over the last decade to exploit the observation that mixed\nvariational problems can be posed on a Hilbert complex, and Galerkin-type mixed\nmethods can then be obtained by solving finite-dimensional subcomplex problems.\nChen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and\noptimality of an adaptive mixed finite element method using Raviart-Thomas or\nBrezzi-Douglas-Marini elements for Poisson's equation on contractible domains\nin two dimensions, which can be viewed as a boundary problem on the de Rham\ncomplex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371)\ndeveloped fundamental tools for a posteriori analysis on the de Rham complex.\nIn this paper, we use tools in FEEC to construct convergence and complexity\nresults on domains with general topology and spatial dimension. In particular,\nwe construct a reliable and efficient error estimator and a sharper\nquasi-orthogonality result using a novel technique. Without marking for data\noscillation, our adaptive method is a contraction with respect to a total error\nincorporating the error estimator and data oscillation.\n