Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method

In this work we study the residual-free bubbles (RFB) finite element method for solving second order elliptic equations with rapidly varying coefficients. The RFB technique is closely related to both the multiscale finite element method (MsFEM) introduced by Hou, Wu, and Cai [Math. Comp., 68 (1999), pp. 913--943] and the upscaling procedures which are very common in the engineering literature for solving this kind of partial differential equation. We also introduce a variation of the RFB method, based on macrobubbles and referred to as the residual-free macrobubbles (RFMB) method, which gives more accurate numerical solutions. In the case of periodic coefficients we are able to prove a priori error estimates for the methods. Eventually, we test the numerical methods on model problems.