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 a...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.展开更多
Over the last ten years, Finite Element Exterior Calculus （FEEC） has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analy...Over the last ten years, Finite Element Exterior Calculus （FEEC） has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Hoist and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations oil Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Hoist, and Zhu extended FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces （spaces of Banach-space valued parametrized curves）. Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst- Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through tile use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in tile flat （non-criminal） setting. Some numerical examples are also presented.展开更多
文摘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.
文摘Over the last ten years, Finite Element Exterior Calculus （FEEC） has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Hoist and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations oil Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Hoist, and Zhu extended FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces （spaces of Banach-space valued parametrized curves）. Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst- Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through tile use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in tile flat （non-criminal） setting. Some numerical examples are also presented.
