Localization of elliptic multiscale problems

A rigorous Multiscale Method for semi-linear elliptic problems

In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of o...

Multiscale mass conservative domain decomposition preconditioners for elliptic problems on irregular grids

Multiscale methods can in many cases be viewed as special types of domain decomposition preconditioners. The localisation approximations introduced within the multiscale framework are dependent upon both the heterogeneity of the reservoir and the structure of the computational grid. While previous works on multiscale control volume methods have focused on heterogeneous elliptic problems on regu...

A Framework for Adaptive Multiscale Methods for Elliptic Problems

We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorithms for solving problems with no clear scale separation. We present numerical simulations demonstrating the...

Multiscale Methods for Elliptic Problems

A Newton-scheme Framework for Multiscale Methods for Nonlinear Elliptic Homogenization Problems∗

In this contribution, we present a very general framework for formulating multiscale methods for nonlinear elliptic homogenization problems. The framework is based on a very general coupling of one macroscopic equation with several localized fine-scale problems. In particular, we recover the Heterogeneous Multiscale Method (HMM), the Multiscale Finite Element Method (MsFEM) and the Variational ...