A weak Galerkin finite element method for the stokes equations

• Published in: Advances in computational mathematics Vol. 42; no. 1; pp. 155 - 174
• Main Authors: Wang, Junping, Ye, Xiu
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2016
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Visualization; 65N15; Secondary; The stokes equations; 35J50; 35B45; Polyhedral meshes; Primary; Weak Galerkin; Finite element methods; 76D07; 65N30
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-015-9415-2

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥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.