Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

Numerical Approximations of a Second Order Elliptic Equation with Highly Oscillating Coefficients

Numerical approximations are considered for the solutions of a class of elliptic equations in divergence form in the case of periodic homogenization. Specifically, we consider boundary value problems of the form

where is a bounded domain in with polygonal boundary, f is a function in , and where with is Y-periodic and satisfies

where are constants. denotes the parameter involved in the periodic homogenization.

It is proposed to use both conforming and mixed finite elements to solve the problem . Error estimates depending on and h (mesh size) are established for each method. It's shown that certain mixed finite element methods are more accurate and robust than conforming finite element method for the class of problems studied.

Numerical simulations for one-dimensional problem comparing the approximations obtained by these methods to the approximation of the homogenized problem are presented.