3:30 pm Wednesday, December 14, 2011
ICES / CVC Seminar : Finite Element Exterior Calculus for Evolution Problems by
Andrew Gillette [mail] (UCSD) in ACES 6.304
Finite element exterior calculus (FEEC) analyzes mixed variational problems and their numerical approximation properties using the ideas and tools of Hilbert complexes. This framework, developed by Arnold, Falk and Winther for elliptic and elasticity problems [1], has been extended by Holst and Stern [2,3] to include variational crimes, allowing applications to linear and nonlinear geometric elliptic partial differential equations and surface finite element problems. In this talk, I will present a new extension of FEEC to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Using the framework of Bochner spaces and their associated norms, I will show how FEEC can be applied to establish a priori error estimates for Galerkin finite element approximation of parabolic problems. This approach recovers the results of Thomee [4] for two-dimensional domains and the lowest-order mixed method as a special case, effectively extending these results to arbitrary spatial dimension and an entire family of mixed methods. I will also discuss how the Holst and Stern results aid in novel analysis and error estimates of non-linear evolution problems and evolution problems involving variational crimes. This is joint work with Michael Holst. [1] Arnold, Falk, and Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (2010), 281--354 [2] Holst and Stern, Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces, arXiv:1005.4455. [3] Holst and Stern, Semilinear mixed problems on Hilbert complexes and their numerical approximation, Found. Comput. Math., in press, 2010. [4] Thomee, Galerkin finite element methods for parabolic problems, Springer Verlag, 2006.
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