Exponential Convergence in a Galerkin Least Squares hp-FEM for Stokes Flow

Abstract

A stabilized hp-Finite Element Method (FEM) of Galerkin Least Squares (GLS) type is analyzed for the Stokes equations in polygonal domains. Contrary to the standard Galerkin FEM, this method admits equal-order interpolation in the velocity and the pressure, which is very attractive from an implementational point of view. In conjunction with geometrically refined meshes and linearly increasing approximation orders it is shown that thehp-GLSFEM leads to exponential rates of convergence for solutions exhibiting singularities near corners. To obtain this result a novel hp-interpolant is constructed that approximates pressure functions in certain weighted Sobolev spaces in an $H^1$-conforming way and at exponential rates of convergence on geometric meshes.