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Zbl 1145.76015
Bochev, Pavel B.; Dohrmann, Clark R.; Gunzburger, Max D.
Stabilization of low-order mixed finite elements for the Stokes equations. (English)
[J] SIAM J. Numer. Anal. 44, No. 1, 82-101 (2006). ISSN 0036-1429; ISSN 1095-7170/e

Summary: We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

MSC 2000:: *76D07 Stokes flows
65N30 Finite numerical methods (BVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
76M10 Finite element methods

Keywords: Stokes equations; stabilized mixed methods; equal-order interpolation; inf-sup condition

Cited in: Zbl 1224.65267 Zbl 1177.76191

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