Generalized FEM for Homogenization Problems

Abstract

We introduce the concept of generalized Finite Element Method (gFEM) for the numerical treatment of homogenization problems. These problems are characterized by highly oscillatory periodic (or patchwise periodic) pattern in the coefficients of the differential equation and their solutions exhibit a multiple scale behavior: a macroscopic behavior superposed with local characteristics at micro length scales. The gFEM is based on two-scale FE spaces that are obtained by augmenting the standard polynomial FE spaces with problem dependent, non-polynomial micro shape functions that reflect the oscillatory behavior of the solution. Our choice of micro-scale shape functions is motivated by a new class of representations formulas for the solutions on unbounded domains, which are based on an assumption of scale separation and generalized Fourier inversion integrals. Under analiticity assumptions on the input data, gFEM converges robustly and exponentially. The micro-shape functions obtained from the representation formula are solutions to suitable unit-cell problems. In the gFEM, they are obtained in the start-up phase of the calculations by solving numerically these unit-cell problems. Different choices of the micro shape functions are possible within the framework of the gFEM and we also analyze gFEM based on micro shape functions obtained from the theory of Bloch waves. Numerical results for one-dimensional and two-dimensional problems corroborate our theoretical results.