Numerical analysis for a new non-conforming linear finite element on quadrilaterals

Starting with a short introduction of a new non-conforming linear quadrilateral P˜1-finite element which has been recently proposed by Park [A study on locking phenomena in finite element methods, Ph.D. Thesis, Seoul National University, February 2002] and Park and Sheen [P1P1-Nonconforming quadrilateral finite element methods for second-order elliptic problems, SIAM J. Numer. Anal. 41(2) (2003) 624–640], we examine in detail the numerical behaviour of this element with special emphasis on the treatment of Dirichlet boundary conditions, efficient matrix assembly and solver aspects. Furthermore, we compare the numerical characteristics of P˜1 with other low-order finite elements, also regarding its use for the incompressible Navier–Stokes equations. Several test examples show the efficiency and reliability of the proposed methods for elliptic second-order problems.