Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces
University of California, San Diego（加利福尼亚大学圣迭戈分校）
Michael Holst, Ari Stern
备注：
27 pages; v2: minor revisions, most notably a new theorem (3.7) on existence of bounded cochain projections and a worked example (4.6) for the Hodge Laplacian on 2-D surfaces
Arnold、Falk和Winther最近的一篇论文[Bull AMS, 47 (2010)]表明，一大类混合有限元方法可以自然地表述在希尔伯特复形上，其中使用类似Galerkin的方法，在有限维子复形上求解变分问题。在一个看似无关的研究方向上，Dziuk[Lect Notes in Math, vol 1357 (1988)]分析了一类用于光滑二维曲面上Laplace-Beltrami方程的节点有限元，该曲面由分段线性三角剖分逼近；Demlow后来将这一分析[SIAM J Numer Anal, 47 (2009)]推广到三维曲面以及高阶曲面逼近。在本文中，我们将这些研究方向结合起来，首先为抽象希尔伯特复形中的变分犯罪建立一个分析框架，然后将该抽象框架应用于超曲面上的有限元外微积分。我们的框架将Arnold、Falk和Winther的工作推广到违反其子复形假设的问题，允许将有限元外微积分扩展到近似域，最显著的是近似流形上的Hodge-de Rham复形。作为后者的应用，我们恢复了Dziuk和Demlow关于二维和三维曲面的先验估计，表明曲面有限元方法可以完全在这个抽象框架内分析。此外，我们的结果极大地推广了这些早期估计，将它们从Laplace-Beltrami的节点有限元扩展到Hodge Laplacian的混合有限元，并从二维和三维超曲面扩展到任意维超曲面。通过结合微分几何和泛函分析的一般工具开发这一分析框架，我们得到了对曲面有限元方法更几何的分析，使得主要结果更加透明。
英文摘要：
A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direction, Dziuk [Lect Notes in Math, vol 1357 (1988)] analyzed a class of nodal finite elements for the Laplace-Beltrami equation on smooth 2-surfaces approximated by a piecewise-linear triangulation; Demlow later extended this analysis [SIAM J Numer Anal, 47 (2009)] to 3-surfaces, as well as to higher-order surface approximation. In this article, we bring these lines of research together, first developing a framework for the analysis of variational crimes in abstract Hilbert complexes, and then applying this abstract framework to the setting of finite element exterior calculus on hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge-de Rham complex on approximate manifolds. As an application of the latter, we recover Dziuk's and Demlow's a priori estimates for 2- and 3-surfaces, demonstrating that surface finite element methods can be analyzed completely within this abstract framework. Moreover, our results generalize these earlier estimates dramatically, extending them from nodal finite elements for Laplace-Beltrami to mixed finite elements for the Hodge Laplacian, and from 2- and 3-dimensional hypersurfaces to those of arbitrary dimension. By developing this analytical framework using a combination of general tools from differential geometry and functional analysis, we are led to a more geometric analysis of surface finite element methods, whereby the main results become more transparent.
Local coderivatives and approximation of Hodge Laplace problems
The Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校计算科学与工程研究所）；Department of Mathematics, University of Oslo（奥斯陆大学数学系）
