2015/09/18 by Michael Holst, Holst, Michael, Christopher Tiee +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1509.05524
openalex publication_date 2015/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Over the last ten years, the Finite Element Exterior Calculus (FEEC) has been\ndeveloped as a general framework for linear mixed variational problems, their\nnumerical approximation by mixed methods, and their error analysis. The basic\napproach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal\narticles in 2006 and 2010, interprets these problems in the setting of Hilbert\ncomplexes, leading to a more general and complete understanding. Over the last\nfive years, the FEEC framework has been extended to a broader set of problems.\nOne such extension, due to Holst and Stern in 2012, was to problems with\nvariational crimes, allowing for the analysis and numerical approximation of\nlinear and geometric elliptic partial differential equations on Riemannian\nmanifolds of arbitrary spatial dimension. Their results substantially\ngeneralize the existing surface finite element approximation theory in several\nrespects. In 2014, Gillette, Holst, and Zhu extended the FEEC in another\ndirection, namely to parabolic and hyperbolic evolution systems by combining\nthe FEEC framework for elliptic operators with classical approaches for\nparabolic and hyperbolic operators, by viewing solutions to the evolution\nproblem as lying in Bochner spaces (spaces of Banach-space valued parametrized\ncurves). Related work on developing an FEEC theory for parabolic evolution\nproblems has also been done independently by Arnold and Chen. In this article,\nwe extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems\non Riemannian manifolds, through the use of framework developed by Holst and\nStern for analyzing variational crimes. We establish a priori error estimates\nthat reduce to the results from earlier work in the flat (non-criminal)\nsetting. Some numerical examples are also presented.\n