Hybridization and postprocessing in finite element exterior calculus

Abstract

We hybridize the methods of finite element exterior calculus for the Hodge–Laplace problem on differential k k -forms in R n \mathbb {R}^n . In the cases k = 0 k=0 and k = n k=n , we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for 0 > k > n 0>k>n , we obtain new hybrid finite element methods, including methods for the vector Poisson equation in n = 2 n=2 and n = 3 n=3 dimensions. We also generalize Stenberg postprocessing [RAIRO Modél. Math. Anal. Numér. 25 (1991), pp. 151–167] from k = n k=n to arbitrary k k , proving new superconvergence estimates. Finally, we discuss how this hybridization framework may be extended to include nonconforming and hybridizable discontinuous Galerkin methods.