A rigorous Multiscale Method for semi-linear elliptic problems

A Localized Orthogonal Decomposition Method for Semi-linear Elliptic Problems

In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small...

Reduced order modeling techniques for numerical homogenization methods applied to linear and nonlinear multiscale problems

The characteristic of effective properties of physical processes in heterogeneous media is a basic modeling and computational problem for many applications. As standard numerical discretization of such multiscale problems (e.g. with classical finite element method (FEM)) is often computationally prohibitive, there is a need for a novel computational algorithm able to capture the effective behav...

Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case)

A Novel Method for Solving Multiscale Elliptic Problems with Randomly Perturbed Data

We propose a method for efficient solution of elliptic problems with multiscale features and randomly perturbed coefficients. We use the multiscale finite element method (MsFEM) as a starting point and derive an algorithm for solving a large number of multiscale problems in parallel. The method is intended to be used within a Monte Carlo framework where solutions corresponding to samples of the...

Analysis of the finite element heterogeneous multiscale method for nonlinear elliptic homogenization problems

An analysis of the finite element heterogeneous multiscale method for a class of quasilinear elliptic homogenization problems of nonmonotone type is proposed. We obtain optimal convergence results for dimension d ≤ 3. Our results, which also take into account the microscale discretization, are valid for both simplicial and quadrilateral finite elements. Optimal a-priori error estimates are obta...