Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

Y. Efendiev, J. Galvis, S. Ki Kang, R.D. Lazarov

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results. © 2012 Global-Science Press.
Original languageEnglish (US)
Pages (from-to)359-383
Number of pages25
JournalNumerical Mathematics: Theory, Methods and Applications
Issue number3
StatePublished - Jun 2012
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The research of Y. Efendiev, J. Galvis, and R. Lazarov has been supported in parts by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). S. Ki Kang and R. Lazarov are also supported in part by the award made by NSF DMS-1016525. S. K. Kang is grateful to Fraunhofer Institute for Industrial Mathematics (ITWM) for hosting her visit in the Spring 2011.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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