A Weak Galerkin Finite Element Method for the Stokes Equations

Abstract

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree $k\ge 1$ for the velocity and polynomials of degree $k-1$ for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree $k-1$ on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.

16 pages

Cited by in corpus (15)

• A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations
• Weak Galerkin Finite Element Methods for the Biharmonic Equation on Polytopal Meshes
• A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations
• A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form
• A Weak Galerkin Finite Element Method for the Maxwell Equations
• A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation
• Analysis of a family of HDG methods for second order elliptic problems
• A Hybridized Formulation for the Weak Galerkin Mixed Finite Element Method
• An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes
• A Discrete Divergence-Free Weak Galerkin Finite Element Method for the Stokes Equations
• Discretization of div-curl Systems by Weak Galerkin Finite Element Methods on Polyhedral Partitions
• New Discretization Schemes for Time-Harmonic Maxwell Equations by Weak Galerkin Finite Element Methods
• A Weak Galerkin Finite Element Method for A Type of Fourth Order Problem Arising From Fluorescence Tomography
• Weak Galerkin method for the coupled Darcy-Stokes flow
• A Weak Galerkin Finite Element Scheme for solving the stationary Stokes Equations