A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian

TL;DR

This paper presents a new nonconforming hybrid finite element method for the 2D vector Laplacian that ensures consistency with a primal variational principle, supports high-order elements, and efficiently handles corner singularities.

Contribution

It introduces a novel nonconforming hybrid finite element approach based on a primal variational principle, with stability and high-order convergence for domains with corner singularities.

Findings

Achieves higher-order convergence under regularity assumptions.

Recovers the $P_1$-nonconforming method at lowest order.

Uses weighted Sobolev spaces for analysis of corner singularities.

Abstract

We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573-595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the $P_1$-nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509-533], and we 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, for domains admitting corner singularities.