FINITE ELEMENT EXTERIOR CALCULUS FOR EVOLUTION PROBLEMS  

作　　者：Andrew Gillette Michael Hoist Yunrong Zhu 

机构地区：[1]Department of Mathematics, University of Arizona, Tucson AZ 85721 [2]Department of Mathematics, University of California San Diego, La Jolla CA 92093 [3]Department of Mathematics & Statistics, Idaho State University, Pocatello 119 83209

出　　处：《Journal of Computational Mathematics》2017年第2期187-212,共26页计算数学（英文）

摘　　要：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 man- ifolds 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 Thom^e 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 re- sults 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.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 man- ifolds 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 Thom^e 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 re- sults 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.

关 键 词：FEEC Elliptic equations Evolution equations Nonlinear equations Approx-imation theory Nonlinear approximation Inf-sup conditions A priori estimates. 

分 类 号：O[理学]