Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces
This paper brings together three distinct ideas that have influenced in separate ways the same area, namely geometric finite element methods for elliptic partial differential equations. The first is that of a variational crime coming from the nonconformity of the approximating space. The second is that of surface finite elements. The last is that of posing a mixed variational problem on a Hilbert complex, a differential complex of Hilbert spaces, that has culminated in the powerful theory of finite element exterior calculus developed by Arnold, Falk, and Winther. The authors first develop a framework for the analysis of variational crimes in abstract Hilbert complexes, and then apply this abstract framework to the setting of finite element exterior calculus on hypersurfaces. This 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. This extension can be applied to recover and dramatically generalize Dziuk's and Demlow's a priori estimates for 2- and 3-surfaces.
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