Numerical methods for elliptic partial differential equations with rapidly oscillating coefficients

Abstract

This paper presents two methods for the numerical solution of the classical homogenization problem of elliptic operators with periodically oscillating coefficients. The numerical solution of such problems is difficult because of the presence of rapidly oscillating coefficients. The first method based on the classical one which consists of the homogenized solution, the first- and second-order correctors, whereas the second one is based on the Bloch wave approach. Further, for the calculation of the homogenized coefficients and some auxiliary functions involved in this method, we applied both methods and compared their accuracies. The Bloch approximation consists in determining an oscillating integral, numerically. The Bloch method provides a better approximation to the exact solution than the classical first-order corrector term in the smooth coefficients case. Moreover, we provided Taylor approximations for the Bloch approximation function and implemented it numerically. In order to show the efficiency of these methods, exhaustive numerical examples in both one and two-dimensional cases are presented.