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

Abdulle, A.

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].

MSC:

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

Keywords:

finite element methods based on the heterogeneous multiscale method; multiscale partial differential equations; porous media flow; biology; material sciences

Citations:

Zbl 1093.35012