Weak Galerkin method for the Stokes equations with damping

Abstract

In this paper, we introduce the weak Galerkin (WG) finite element method for the Stokes equations with damping. We establish the WG numerical scheme on general meshes and prove the well-posedness of the scheme. Optimal error estimates for the velocity and pressure are derived. Furthermore, in order to accelerate the WG algorithm, we present a two-level method and give the corresponding error estimates. Finally, some numerical examples are reported to validate the theoretical analysis.

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

“…The WG method is first introduced by Wang and Ye [17,18] for the second-order elliptic equations, and a stabilizer term is added to WG-FEM in order to enforce the connection of discontinuous functions across element boundaries [10,11]. Then the WG method finds applications in diverse areas including elliptic equations [9,27], parabolic equations [30,31,34,35], second-order linear wave equation [6], reaction-diffusion equations [8], Stokes equations [13,16,32], Maxwell equations [12,14,20], biharmonic equation [33], Cahn-Hilliard-Cook equation [5], stochastic parabolic equations [36,37], eigenvalue problems [28,29], and so on. Nevertheless, the stabilizer makes the finite element formulations and programming complex.…”

Superconvergence analysis of the bilinear‐constant scheme for two‐dimensional incompressible convective Brinkman–Forchheimer equations

“…Much work has been devoted to theoretical analysis of partial differential equations with the term $$$ \alpha {\left|u\right|}^{r-2}u $$$ $$$ \left(r\ge 2\right) $$$, which was called the damping term or source term in [5‐8]. As for the Stokes/Navier–Stokes equations with damping, we refer to [1, 9‐18] and the references therein. In particular, the Taylor‐Hood mixed finite element method was used to approximate the two‐dimensional convective Brinkman–Forchheimer equations in [1] and optimal error estimates were established.…”

“…The consistency and stabilization of the numerical schemes were proved and optimal error estimates were derived. The weak Galerkin finite element method was investigated for the Stokes equations with damping on general meshes in [14]. The well-posedness of the scheme was discussed and optimal error estimates for the velocity and pressure were derived.…”

A three‐step Oseen‐linearized finite element method for incompressible flows with damping

“…The basic idea of standard two-level method is that solve a small nonlinear problem on a coarse grid to capture the "large eddies" and solve a linear problem on a fine grid to capture the fine structures. Motivated by the work of Xu, some standard two-level finite element methods were proposed for the Stokes or Navier-Stokes problem with damping [25][26][27], where the method studied in [27] is based on weak Galerkin finite element discretization. In [28,29], multi-level stabilized finite element methods and two-level defect-correction stabilized methods using the lowest equal-order elements were studied, respectively.…”