9,960 research outputs found
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
This paper presents two new approaches for finding the homogenized
coefficients of multiscale elliptic PDEs. Standard approaches for computing the
homogenized coefficients suffer from the so-called resonance error, originating
from a mismatch between the true and the computational boundary conditions. Our
new methods, based on solutions of parabolic and elliptic cell-problems, result
in an exponential decay of the resonance error
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
A machine learning approach for efficient uncertainty quantification using multiscale methods
Several multiscale methods account for sub-grid scale features using coarse
scale basis functions. For example, in the Multiscale Finite Volume method the
coarse scale basis functions are obtained by solving a set of local problems
over dual-grid cells. We introduce a data-driven approach for the estimation of
these coarse scale basis functions. Specifically, we employ a neural network
predictor fitted using a set of solution samples from which it learns to
generate subsequent basis functions at a lower computational cost than solving
the local problems. The computational advantage of this approach is realized
for uncertainty quantification tasks where a large number of realizations has
to be evaluated. We attribute the ability to learn these basis functions to the
modularity of the local problems and the redundancy of the permeability patches
between samples. The proposed method is evaluated on elliptic problems yielding
very promising results.Comment: Journal of Computational Physics (2017
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
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
- …