Local Finite Element Approximation of Sobolev Differential Forms

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Cl\'ement interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.Comment: 22 pages. Comments welcom

Partial expansion of a Lipschitz domain and some applications

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated

Smoothed projections over manifolds in finite element exterior calculus

We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on manifolds. The commuting projections use localized mollification operators, building upon a classical construction by de Rham. These projections are uniformly bounded on Lebesgue spaces of differential forms and map onto intrinsic finite element spaces defined with respect to an intrinsic smooth triangulation of the manifold. We analyze the Galerkin approximation error. Since practical computations use extrinsic finite element methods over approximate computational manifolds, we also analyze the geometric error incurred.Comment: Submitted. 31 page

Convergence and Optimality of Adaptive Mixed Methods on Surfaces

In a 1988 article, Dziuk introduced a nodal finite element method for the Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear triangulation, initiating a line of research into surface finite element methods (SFEM). Demlow and Dziuk built on the original results, introducing an adaptive method for problems on 2-surfaces, and Demlow later extended the a priori theory to 3-surfaces and higher order elements. In a separate line of research, the Finite Element Exterior Calculus (FEEC) framework has been developed over the last decade by Arnold, Falk and Winther and others as a way to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can be obtained by solving finite dimensional subproblems. In 2011, Holst and Stern merged these two lines of research by developing a framework for variational crimes in abstract Hilbert complexes, allowing for application of the FEEC framework to problems that violate the subcomplex assumption of Arnold, Falk and Winther. When applied to Euclidean hypersurfaces, this new framework recovers the original a priori results and extends the theory to problems posed on surfaces of arbitrary dimensions. In yet another seemingly distinct line of research, Holst, Mihalik and Szypowski developed a convergence theory for a specific class of adaptive problems in the FEEC framework. Here, we bring these ideas together, showing convergence and optimality of an adaptive finite element method for the mixed formulation of the Hodge Laplacian on hypersurfaces.Comment: 22 pages, no figures. arXiv admin note: substantial text overlap with arXiv:1306.188