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.

翻译：我们对二维矢量 Laplacian 采用了一种不兼容的混合有限元素方法,其基础是初步变异原则,已知其方法不一致。使用与稳定混合不连续加列金(HDG)方法相似的处罚术语,确保了一致性,由于Brenner、Li和Sung[Math.comp. 76(2007),pp.573-595],我们仔细选择了惩罚参数。我们的方法包含任意高顺序元素,与HDG方法一样,可高效使用静态凝聚。最低顺序案例恢复了Brenner、Cui、Li和Sung[Numer. Math.,109(2008),pp.509-533]的不兼容方法。我们表明,在适当的常规假设下实现了较高顺序的趋同。分析对Kondrat'ev的加权索博列夫空间组合进行了新颖的利用,因为Kondrat'ev,用于承认角点的域。