P1-Nonconforming Finite Elements on Triangulations into Triangles and Quadrilaterals

The $P_1$-nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis for the combination of the Park-Sheen and the Crouzeix-Raviart nonconforming finite element methods is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions.