Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model

Abstract

In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (P1-P1-P1) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms.