A new approximate method for second order elliptic problems with rapidly oscillating coefficients based on the method of multiscale asymptotic expansions

Abstract

In this paper, we consider second order elliptic problems with rapidly oscillating coefficients. On basis of [O.A. Oleinik, A.S. Shamaev, G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992; Wen-ming He, Jun-zhi Cui, A pointwise estimate on the 1-order approximation of G ε x 0 , IMA J. Appl. Math. 70 (2005) 241-269] we propose a new approximate method to solve these problems. Of course, we present its error estimate.

Finite volume discretization of a class of parabolic equations with discontinuous and oscillating coefficients

“…Proposition 2.3 Let T = ( I i ) i=0,...,N −1 be an admissible mesh of (0, 1) in the sens of Definition 2.1, such as the discontinuities of k ε and c ε coincide with the interfaces of the mesh. Then the implicit scheme (21), where the matrix A ε is given by (17) is L ∞ stable.…”

General decay of energy for a viscoelastic equation with nonlinear damping

“…The blow-up result to the solutions with positive initial energy for this problem was established under suitable assumptions on g; r and p. For more related results about the existence, finite time blow-up and decay estimate, we refer to the reader to [1,2,5,6,8,11,17].…”

The Effect of Numerical Integration on the Finite Element Approximation of a Second Order Elliptic Equation with Highly Oscillating Coefficients

“…The purpose of this paper is to show that even when h ε is an integer the numerical solution with numerical integration effects can be correct in some cases depending on the shape of coefficients. The numerical approximation partial differential equations with highly oscillating coefficients has been a problem of interest for many years and many methods have been developed (see, e.g., [1], [6], [7], [10], [11], [13] and the bibliographies therein). The paper is organized as follows.…”