The Effect of Numerical Integration on the Finite Element Approximation of a Second Order Elliptic Equation with Highly Oscillating Coefficients
Bienvenu Ondami · Journal of Interpolation and Approximation in Scientific Computing · 2015
In this paper, we have studied the effect of numerical integration on the Finite Element Method (FEM) based on the usual Ritz approximation using continuous piecewise linear functions, in the context of a class of second order elliptic boundary value problems with highly periodically oscillating coefficients. An error estimate depending on $\varepsilon$ the parameter involved in the periodic homogenization and $h$ the mesh size is established. Numerical results for one dimensional problem are presented. It is shown that when $\frac{h}{\varepsilon}$ is a positive integer then the method gives different results depending on the shape of the coefficients and the numerical integration. Specifically we obtain perfectly correct results in some cases and completely false in other cases.