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Mixed multiscale methods for heterogeneous elliptic problems. (English) Zbl 1248.65119

Graham, Ivan G. (ed.) et al., Numerical analysis of multiscale problems. Selected papers based on the presentations at the 91st London Mathematical Society symposium, Durham, UK, July 5–15, 2010. Berlin: Springer (ISBN 978-3-642-22060-9/hbk; 978-3-642-22061-6/ebook). Lecture Notes in Computational Science and Engineering 83, 243-283 (2012).
Summary: We consider a second order elliptic problem written in mixed form, i.e., as a system of two first order equations. Such problems arise in many contexts, including flow in porous media. The coefficient in the elliptic problem (the permeability of the porous medium) is assumed to be spatially heterogeneous. The emphasis here is on accurate approximation of the solution with respect to the scale of variation in this coefficient. Homogenization and upscaling techniques alone are generally inadequate for this problem. As an alternative, multiscale numerical methods have been developed. They can be viewed in one of three equivalent frameworks: as a Galerkin or finite element method with nonpolynomial basis functions, as a variational multiscale method with standard finite elements, or as a domain decomposition method with restricted degrees of freedom on the interfaces. We treat each case, and discuss the advantages of the approach for devising effective local multiscale methods. Included is recent work on methods that incorporate information from homogenization theory and effective domain decomposition methods.
For the entire collection see [Zbl 1234.65007].

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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