The role of numerical integration in numerical homogenization

Finite elements methods (FEMs) with numerical integration play a central role in numerical homogenization methods for partial differential equations with multiple scales, as the effective data in a homogenization problem can only be recovered from a microscopic solver at a finite number of points in the computational domain. In a multiscale framework the convergence of a FEM with numerical integration applied to the effective (homogenized) problem guarantees that the so-called macroscopic solver is consistent and convergent. Convergence results for FEM with numerical integration are however scarce in the literature and need often to be derived as a first step to analyze a numerical homogenization method for a given problem. In this paper we review and explain the main ideas in deriving convergence results for FEM with numerical integration for linear and nonlinear elliptic problems and explain the role of these methods in numerical homogenization.