Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Schöberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincaré-Friedrichs-type inequalities will be studied in a subsequent contribution.2000 Mathematics Subject Classification. 65N30, 58A12. Key words and phrases. discrete distributional differential form, finite element exterior calculus, finite element method, harmonic form, a posteriori error estimation.

Abstract. Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve homogeneous boundary conditions along a fixed boundary part, and satisfy uniform bounds for shape-regular families of triangulations and bounded polynomial degree. The existence of such projections implies stability and quasi-optimal convergence of mixed finite element methods for the Hodge Laplace equation with mixed boundary conditions. In addition, we prove the density of smooth differential forms in Sobolev spaces of differential forms over weakly Lipschitz domains with partial boundary conditions.

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The Stokes equation posed on surfaces is important in some physical models, but its numerical solution poses several challenges not encountered in the corresponding Euclidean setting. These include the fact that the velocity vector should be tangent to the given surface and the possible presence of degenerate modes (Killing fields) in the solution. We analyze a surface finite element method which provides solutions to these challenges. We consider an interior penalty method based on the well-known Brezzi-Douglas-Marini H(div)-conforming finite element space. The resulting spaces are tangential to the surface, but require penalization of jumps across element interfaces in order to weakly maintain H 1 conformity of the velocity field. In addition our method exactly satisfies the incompressibility constraint in the surface Stokes problem. Secondly, we give a method which robustly filters Killing fields out of the solution. This problem is complicated by the fact that the dimension of the space of Killing fields may change with small perturbations of the surface. We first approximate the Killing fields via a Stokes eigenvalue problem and then give a method which is asymptotically guaranteed to correctly exclude them from the solution. The properties of our method are rigorously established via an error analysis and illustrated via numerical experiments.

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 Clément 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.