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：本文针对二维向量拉普拉斯算子提出一种非协调混合有限元方法，该方法基于原始变分原理，而传统协调方法在此原理下已知存在不一致性。通过引入与混合化间断伽辽金（HDG）方法稳定性处理类似的惩罚项，并采用Brenner、Li与Sung [Math. Comp., 76 (2007), pp. 573-595] 精心选取的惩罚参数，确保了方法的相容性。本方法适用于任意高阶单元，且与HDG方法类似，可通过静态凝聚实现高效计算。最低阶情形可还原Brenner、Cui、Li与Sung [Numer. Math., 109 (2008), pp. 509-533] 提出的 $P_1$ 非协调方法，同时在适当正则性假设下证明了高阶收敛性。分析过程中创新性地引入了Kondrat'ev提出的加权Sobolev空间族，用以处理存在角奇异性区域。