468 research outputs found
A Multiscale Enrichment Procedure for Nonlinear Monotone Operators
In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis
Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media
In this paper we propose and analyze a preconditioner for a system arising
from a finite element approximation of second order elliptic problems
describing processes in highly het- erogeneous media. Our approach uses the
technique of multilevel methods and the recently proposed preconditioner based
on additive Schur complement approximation by J. Kraus (see [8]). The main
results are the design and a theoretical and numerical justification of an
iterative method for such problems that is robust with respect to the contrast
of the media, defined as the ratio between the maximum and minimum values of
the coefficient (related to the permeability/conductivity).Comment: 28 page
Asymptotic expansions for high-contrast elliptic equations
In this paper, we present a high-order expansion for elliptic equations in
high-contrast media. The background conductivity is taken to be one and we
assume the medium contains high (or low) conductivity inclusions. We derive an
asymptotic expansion with respect to the contrast and provide a procedure to
compute the terms in the expansion. The computation of the expansion does not
depend on the contrast which is important for simulations. The latter allows
avoiding increased mesh resolution around high conductivity features. This work
is partly motivated by our earlier work in \cite{ge09_1} where we design
efficient numerical procedures for solving high-contrast problems. These
multiscale approaches require local solutions and our proposed high-order
expansion can be used to approximate these local solutions inexpensively. In
the case of a large-number of inclusions, the proposed analysis can help to
design localization techniques for computing the terms in the expansion. In the
paper, we present a rigorous analysis of the proposed high-order expansion and
estimate the remainder of it. We consider both high and low conductivity
inclusions
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