Error estimates for two-scale composite finite element approximations of parabolic equations with measure data in time for convex and nonconvex polygonal domains

Abstract

In this exposition we study two-scale composite finite element approximations of parabolic problems with measure data in time for both convex and nonconvex polygonal domains. This research is motivated by the work of Hackbusch and Sauter [Numer. Math., 75 (1997) 447–472] on the composite finite element approximations of elliptic boundary value problems. The main features of the composite finite element method is that, it not only uses minimal dimension of the approximation space but also handle the domain boundary in a flexible and systematic manner, which is very advantageous for domains with complicated geometry. Both spatially semidiscrete and fully discrete approximations of the proposed method are analyzed. In the case of convex domains, we derive error estimate of order O(HLog˜1/2(H/h)+k1/2) in the L2(0,T;L2(Ω))-norm, where H and h denote the coarse-scale and fine-scale mesh size, respectively, and k is the time step. Further, an error estimate of order O(HsLog˜s/2(H/h)+k1/2), 1/2≤s≤1 is shown to hold in the L2(0,T;L2(Ω))-norm for nonconvex domains. Numerical experiment confirms the theoretical findings and reveals the potential of the composite finite element method.