A Multiscale Finite Element Method for an Elliptic Distributed Optimal Control Problem with Rough Coefficients and Control Constraints

We construct and analyze a multiscale finite element method for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for smooth problems and present corroborating numerical results.Comment: 26 page

Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation

We continue the study of the nonconforming multiscale finite element method (Ms- FEM) introduced in 17, 14 for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of O(Є^2/h^2). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems

Multiscale Finite Element Methods for Nonlinear Problems and their Applications

In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities

Reduced basis heterogeneous multiscale methods

Numerical methods for partial differential equations with multiple scales that combine numerical homogenization methods with reduced order modeling techniques are discussed. These numerical methods can be applied to a variety of problems including multiscale nonlinear elliptic and parabolic problems or Stokes flow in heterogenenous media

A method for elliptic problems with high-contrast coefficients.

Lee, Ho Fung.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 75-79).Abstracts in English and Chinese.Chapter 1 --- Upscaling methods for high contrast problems --- p.6Chapter 1.1 --- Review on upscaling methods --- p.7Chapter 1.2 --- Upscaling method with high contrast of the conductivity --- p.11Chapter 2 --- Multiscale finite element methods for high contrast problems --- p.19Chapter 2.1 --- Review on Multiscale finite element methods --- p.20Chapter 2.2 --- Local spectral basis functions --- p.23Chapter 2.3 --- Discussion for MsFEM with spectral multiscale basis functions . --- p.25Chapter 3 --- Elliptic equations in high-contrast heterogeneous media --- p.28Chapter 3.1 --- Preliminaries --- p.29Chapter 3.2 --- Integral representation --- p.32Chapter 3.3 --- The well-posedness of the integral equation --- p.37Chapter 4 --- A numerical approach for the Elliptic equations in high-contrast heterogeneous media --- p.45Chapter 4.1 --- Introduction --- p.46Chapter 4.2 --- A new approach --- p.47Chapter 4.3 --- Discussion of the results --- p.50Chapter 4.4 --- Numerical experiment --- p.51Chapter 5 --- A preconditioner for high contrast problems using reduced-contrast approximations --- p.62Chapter 5.1 --- Reduced-contrast approximations for the solution of elliptic equations --- p.63Chapter 5.2 --- Review on multigrid methods --- p.66Chapter 5.3 --- Preconditioning and numerical experiments --- p.70Bibliography --- p.7