Volume 35, Issue 2
Finite Element Exterior Calculus for Evolution Problems

J. Comp. Math., 35 (2017), pp. 187-212.

Published online: 2017-04

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• Abstract

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263-293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomée for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomée and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

• Keywords

FEEC, Elliptic equations, Evolution equations, Nonlinear equations, Approximation theory, Nonlinear approximation, Inf-sup conditions, $A$ priori estimates.

65M15, 65M20, 65M60.

agillette@math.arizona.edu (Andrew Gillette)

mholst@math.ucsd.edu (Michael Holst)

zhuyunr@isu.edu (Yunrong Zhu)

• BibTex
• RIS
• TXT
@Article{JCM-35-187, author = {Gillette , AndrewHolst , Michael and Zhu , Yunrong}, title = {Finite Element Exterior Calculus for Evolution Problems}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {2}, pages = {187--212}, abstract = {

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263-293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomée for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomée and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1610-m2015-0319}, url = {http://global-sci.org/intro/article_detail/jcm/9769.html} }
TY - JOUR T1 - Finite Element Exterior Calculus for Evolution Problems AU - Gillette , Andrew AU - Holst , Michael AU - Zhu , Yunrong JO - Journal of Computational Mathematics VL - 2 SP - 187 EP - 212 PY - 2017 DA - 2017/04 SN - 35 DO - http://doi.org/10.4208/jcm.1610-m2015-0319 UR - https://global-sci.org/intro/article_detail/jcm/9769.html KW - FEEC, Elliptic equations, Evolution equations, Nonlinear equations, Approximation theory, Nonlinear approximation, Inf-sup conditions, $A$ priori estimates. AB -

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263-293 and 363-387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomée for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomée and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

Andrew Gillette, Michael Holst & Yunrong Zhu. (2020). Finite Element Exterior Calculus for Evolution Problems. Journal of Computational Mathematics. 35 (2). 187-212. doi:10.4208/jcm.1610-m2015-0319
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