Approximation of generalized Stokes problems using dual-mixed finite elements without enrichment

• Published in: International journal for numerical methods in fluids 2011-09, Vol.67 (2), p.247-268
• Main Author: Howell, Jason S.
• Format: Article
• Language: English
• Subjects: Approximation; Computational fluid dynamics; Computational methods in fluid dynamics; dual-mixed method; Exact sciences and technology; Finite element method; Fluid dynamics; Fluid flow; Fundamental areas of phenomenology (including applications); generalized Stokes problem; Joints; Laminar flows; Low-reynolds-number (creeping) flows; Mathematical analysis; Mathematical models; Physics; pseudostress; Raviart-Thomas; Stokes law (fluid mechanics); Stokes problem; twofold saddle point problem
• ISSN: 0271-2091; 1097-0363; 1097-0363
• DOI: 10.1002/fld.2356

In this work a finite element method for a dual‐mixed approximation of generalized Stokes problems in two or three space dimensions is studied. A variational formulation of the generalized Stokes problems is accomplished through the introduction of the pseudostress and the trace‐free velocity gradient as unknowns, yielding a twofold saddle point problem. The method avoids the explicit computation of the pressure, which can be recovered through a simple post‐processing technique. Compared with an existing approach for the same problem, the method presented here reduces the global number of degrees of freedom by up to one‐third in two space dimensions. The method presented here also represents a connection between existing dual‐mixed and pseudostress methods for Stokes problems. Existence, uniqueness, and error results for the generalized Stokes problems are given, and numerical experiments that illustrate the theoretical results are presented. Copyright © 2010 John Wiley & Sons, Ltd.