Preconditioning Heterogeneous $\boldsymbol{H}(\mathrm{div})$ Problems by Additive Schur Complement Approximation and Applications

Abstract

In this paper we propose and analyze a preconditioner for a system arising from a mixed finite element approximation of second-order elliptic problems describing processes in highly heterogeneous media. Our approach uses the technique of multilevel methods (see, e.g., [P. Vassilevski, Multilevel Block Factorization Preconditioners: Matrix-Based Analysis and Algorithms for Solving Finite Element Equations, Springer, New York, 2008]) and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus [SIAM J. Sci. Comput., 34 (2012), pp. A2872--A2895]. The main results are the design, study, and numerical justification of iterative algorithms for these problems that are robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient of the problem. Numerical tests provide experimental evidence for the high quality of the preconditioner and its desired robustness with respect to the material contrast. Such resu...