Core Concepts
The authors introduce a nonconforming hybrid finite element method for solving the two-dimensional vector Laplacian problem, which ensures consistency using penalty terms similar to those used in hybridizable discontinuous Galerkin (HDG) methods.
Abstract
The key highlights and insights of the content are:
The authors present a three-field primal hybridization of the nonconforming primal finite element method for the two-dimensional vector Laplacian problem. This method accommodates elements of arbitrarily high order and can be implemented efficiently using static condensation.
The lowest-order case of the proposed method recovers the P1-nonconforming method of Brenner et al., and the authors show that higher-order convergence is achieved under appropriate regularity assumptions.
The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, to handle domains admitting corner singularities. This allows the authors to obtain error estimates without imposing mesh-grading conditions required by previous methods.
The authors establish the well-posedness of the hybrid method and demonstrate the equivalence of the three-field, two-field, and one-field formulations. They also discuss the static condensation of the hybrid method, which enables efficient global solvers.
Numerical experiments are presented to confirm the analytically obtained convergence results.