Numerical homogenization is a methodology for the computational solution of multiscale
partial differential equations. It aims at reducing complex large-scale problems to simplified …

This paper reviews standard oversampling strategies as performed in the multiscale finite
element method (MsFEM). Common to those approaches is that the oversampling is …

In this paper, we propose a general approach called Generalized Multiscale Finite Element
Method (GMsFEM) for performing multiscale simulations for problems without scale …

This paper constructs a local generalized finite element basis for elliptic problems with
heterogeneous and highly varying coefficients. The basis functions are solutions of local …

Numerical homogenization, ie, the finite-dimensional approximation of solution spaces of
PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements …

The objective of this book is to introduce the reader to the Localized Orthogonal
Decomposition (LOD) method for solving partial differential equations with multiscale data …

We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/
multiresolution method for PDEs with rough (L^∞) coefficients with rigorous a priori …

In this paper, the generalized finite element method (GFEM) for solving second order elliptic
equations with rough coefficients is studied. New optimal local approximation spaces for …

Numerical homogenization aims to efficiently and accurately approximate the solution space
of an elliptic partial differential operator with arbitrarily rough coefficients in a $ d …

We introduce a new variational method for the numerical homogenization of divergence
form elliptic, parabolic and hyperbolic equations with arbitrary rough (L∞) coefficients. Our …