Finite Element Pressure Stabilizations for Incompressible Flow Problems

Abstract

The simulation of incompressible flow problems with pairs of velocity-pressure finite element spaces that do not satisfy the discrete inf-sup condition requires a so-called pressure stabilization. This chapter provides a survey of available methods which are presented for the Stokes problem to concentrate on the main ideas and to avoid additional difficulties originating from more complicated models. The methods can be divided into residual-based stabilizations and stabilizations that utilize only the pressure. For the first class, a comprehensive numerical analysis is presented, whereas for the second class, the presentation is more concise except for a detailed analysis of a local projection stabilization method. Connections of various pressure stabilizations to inf-sup stable discretizations with velocity spaces enriched by bubble functions are also discussed. Numerical studies compare several of the available pressure stabilizations. Keywords Stokes equations · Discrete inf-sup condition · Pressure-stabilized Petrov-Galerkin (PSPG) method · Galerkin least squares (GLS) method · Douglas-Wang method · Local projection stabilization (LPS) method · Velocity finite element spaces with bubble functions MSC 2010 65N30, 65N12, 65N15