Interior maximum norm estimates for finite element discretizations of the Stokes equations

Abstract

Interior estimates are proved in the L ^∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations.

Keywords:

• Interior error estimates
• Stokes equations
• Mixed finite element method

Keywords:

• 2000 Mathematics Subject Classifications: Primary: 65N15
• Secondary: 65N30