The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. (English) Zbl 1372.65266

Abdulle, Assyr (ed.) et al., Multiple scales problems in biomathematics, mechanics, physics and numerics, CIMPA school, Cape Town, South Africa, August 6–18, 2007. Tokyo: Gakkotosho (ISBN 978-4-7625-0456-3/hbk). GAKUTO International Series. Mathematical Sciences and Applications 31, 133-181 (2009).
Summary: Heterogeneous multiscale methods (HMM) have been introduced by Weinan E and B. Engquist [Commun. Math. Sci. 1, No. 1, 87–132 (2003; Zbl 1093.35012)] as a general methodology for the numerical computation of problems with multiple scales. In this paper we discuss finite element methods based on the HMM for multiscale partial differential equations (PDEs). We give numerous examples of such multiscale problems, including elliptic, parabolic and advection diffusion problems and discuss several applications in areas such as porous media flow, biology and material sciences. A detailed analysis of the methods as well as recent developments are discussed.
For the entire collection see [Zbl 1171.65002].


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76S05 Flows in porous media; filtration; seepage


Zbl 1093.35012
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