Numerical analysis and a-posteriori error control for a new nonconforming linear finite element on quadrilaterals

Starting with a short introduction of the new nonconforming linear quadrilateral P̃1-finite element which has been recently proposed by Park ([13, 14]), we examine in detail the numerical behaviour of this element with special emphasis on the treatment of Dirichlet boundary conditions, efficient matrix assembly, solver aspects and the use as Stokes element in CFD. Furthermore, we compare the numerical characteristics of P̃1 with other low order finite elements. Moreover, we derive a dual weighted residual-based a-posteriori error estimation procedure in the sense of Becker and Rannacher (c.f. [2]) for P̃1. Several test examples show the efficiency and reliability of the proposed method for elliptic 2nd order problems.