Locally stabilized P1P1-nonconforming quadrilateral and hexahedral finite element methods for the Stokes equations

In this paper, we consider locally stabilized   pairs (P1,P1)(P1,P1)-nonconforming quadrilateral and hexahedral finite element methods for the two- and three-dimensional Stokes equations. The stabilization is obtained by adding to the bilinear form the difference between an exact Gaussian quadrature rule for quadratic polynomials and an exact Gaussian quadrature rule for linear polynomials. Optimal error estimates are derived in the energy norm and the L2L2-norm for the velocity and in the L2L2-norm for the pressure. In addition, numerical experiments to confirm the theoretical results are presented. From our numerical results, we observe that the proposed stabilized  (P1,P1)(P1,P1)-nonconforming finite element method shows better performance than the standard method.

► A new finite element method for solving the Stokes problem has been introduced.
► Equal-order finite element spaces for both velocity and pressure are employed.
► Both 2D and 3D quadrilateral and hexahedral meshes are considered.
► Nonconforming elements of the least degrees of freedom, thus the cheapest, are used.
► To enhance stability in calculating pressure, Gauss quadrature is modified.