Reference Work Entry

Encyclopedia of Applied and Computational Mathematics

pp 1066-1074

Date:

Numerical Homogenization

  • Assyr AbdulleAffiliated withMathematics Section, École Polytechnique Fédérale de Lausanne (EPFL)

Synonyms

Multiscale methods for homogenization problems; Representative volume element methods; Upscaling methods

Definition

Numerical homogenization methods are techniques for finding numerical solutions of partial differential equations (PDEs) with rapidly oscillating coefficients (multiple scales). In mathematical analysis, homogenization can be defined as a theory for replacing a PDE with rapidly oscillating coefficients by a PDE with averaged coefficients (an effective PDE) that describes the macroscopic behavior of the original equation. Numerical techniques that are able to approximate the solution of an effective PDE (often unknown in closed form) and local fluctuation of the oscillatory solution without resolving the full oscillatory equation by direct discretization are coined “numerical homogenization methods.” These methods are also called multiscale methods as they typically combine numerical solvers on different scales.

Overview

Homogenization

Consider a general family of PDEs

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