A Mixed Finite Element Method for the Stokes Equations Based on a Weakly Over-Penalized Symmetric Interior Penalty Approach

• Published in: Journal of scientific computing Vol. 58; no. 2; pp. 290 - 307
• Main Authors: Barker, Andrew T., Brenner, Susanne C.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-02-2014; Springer Nature B.V
• Subjects: Adaptive algorithms; Algorithms; Computational Mathematics and Numerical Analysis; Convergence; Decomposition; Finite element analysis; Finite element method; Fluid flow; Hypotheses; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Nodes; Norms; Optimization; Solvers; Theoretical; Velocity; Mixed finite element method; Discontinuous finite element method; Stokes problem; Adaptive algorithm; Weakly over-penalized symmetric interior penalty method; Additive Schwarz preconditioners; Nonconforming meshes
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-013-9733-9

We present a mixed finite element method for the steady-state Stokes equations where the discrete bilinear form for the velocity is obtained by a weakly over-penalized symmetric interior penalty approach. We show that this mixed finite element method is inf-sup stable and has optimal convergence rates in both the energy norm and the
L
2
norm on meshes that can contain hanging nodes. We present numerical experiments illustrating these results, explore a very simple adaptive algorithm that uses meshes with hanging nodes, and introduce a simple but scalable parallel solver for the method.