Efficient linear solvers for incompressible flow simulations using Scott‐Vogelius finite elements1. Issue 4 (5th December 2012)

• Record Type: Journal Article
• Authors: Cousins, Benjamin R.; Borne, Sabine Le; Linke, Alexander; Rebholz, Leo G.; Wang, Zhen
• Is Part Of: Numerical methods for partial differential equations. Volume 29:Issue 4(2013:Jul.)
• Page Start: 1217
• Page End: 1237
• Publication Date: 2012-12-05
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21752
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, to appear) divergence‐free mixed finite elements may have a significantly smaller discretization error than standard nondivergence‐free mixed finite elements. To judge the overall performance of divergence‐free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((<italic>P</italic><sub><italic>k</italic></sub>)<sup><italic>d</italic></sup>, <italic>P</italic><sub><italic>k</italic>‐1</sub><sup><italic>d</italic><italic>i</italic><italic>s</italic><italic>c</italic></sup>) Scott‐Vogelius finite element implementations of the incompressible Navier–Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott‐Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor‐Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and <tex-math notation="LaTeX"><![CDATA[\documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal{H}\end{align*} \end{document}]]></tex-math><inline-graphic xlink:href="ark:/27927/pgg1xr8774g" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> ‐LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013</p> </abstract>