P1-nonconforming quadrilateral finite element methods for second-order elliptic problems

A P-1-nonconforming quadrilateral finite element is introduced for second-order elliptic problems in two dimensions. Unlike the usual quadrilateral nonconforming finite elements, which contain quadratic polynomials or polynomials of degree greater than 2, our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges. One of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements. An optimal rate of convergence is obtained. Also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral. An extension of the P-1-nonconforming element to three dimensions is also given. Finally, several numerical results are reported to confirm the effective nature of the proposed new element.