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Auxiliary space preconditioners for mixed finite element methods. (English)
Bercovier, Michel (ed.) et al., Domain decomposition methods in science and engineering XVIII. Selected papers based on the presentations at the 18th international conference of domain decomposition methods, Jerusalem, Israel, January 12‒17, 2008. Berlin: Springer (ISBN 978-3-642-02676-8/hbk; 978-3-642-04466-3/ebook). Lecture Notes in Computational Science and Engineering 70, 99-109 (2009).
Summary: This paper is devoted to study of an auxiliary spaces preconditioner for ${\bold H}(\text{div})$ systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency and robustness of the algorithms, even in the presence of large coefficient variations. For the mixed formulation of elliptic equations, we use the augmented Lagrange technique to convert the solution of the saddle point problem into the solution of a nearly singular ${\bold H}(\text{div})$ system. Numerical experiments also justify the robustness and efficiency of this scheme.