Stabilization of low-order cross-grid $\mathbf{ P_k Q_l}$ mixed finite elements

MARIA GABRIELA ARMENTANO

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2018 vol. 330 p. 340 - 355

0377-0427

In this paper we analyze a low-order family of mixed finite element methods for the numerical solution of the Stokes problem  and a  second order elliptic problem, in two space dimensions. In these schemes, the pressure is interpolated on a mesh of rectangular elements, while the velocity is approximated on a triangular mesh obtained by dividing eachrectangle into four triangles by its diagonals.For the lowest order $P_1Q_0$, a global spurious pressuremode  is shown to exist and so this element, as $P_1Q_1$ case analyzed in cite{AB}, is unstable. However, following the ideas given in cite{BDG}, a simple stabilization procedure can be applied, when we approximate the solution of the Stokes problem, such that the new $P_1Q_0$ and $P_1Q_1$  methods are unconditionally stable, and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations.Moreover, we analyze the application of our $P_1Q_1$  element to the mixed formulation of the elliptic problem. In this case, by introducing the  modified mixed weak form proposed in cite{BFM}, optimal order of accuracy can be obtained with our stabilized $P_1Q_1$ elements. Numerical results are also presented, which confirmthe existence of the spurious pressure mode for the $P_1Q_0$ element and the excellent stability and accuracy of the new stabilized methods.