On the Convergence Analysis of MsFEM with Oversampling: Interpolation Error

Abstract. In this paper, we investigate the approximation properties of two types of multiscale finite element methods with oversampling as proposed in [Hou and Wu, J. Comput. Phys., 134 (1997), pp. 169–189] and [Efendiev, Hou, and Wu, SIAM J. Numer. Anal., 37 (2000), pp. 888–910] without scale separation. We develop a general interpolation error analysis for elliptic problems with highly oscillatory rough coefficients, under the assumption of the existence of a macroscopic problem with suitable [Formula: see text]-accuracy. The distinct features of the analysis, in the setting of highly oscillatory periodic coefficients, include the following: (i) the analysis is independent of the first-order corrector or the solutions to the cell problems, and thus independent of their regularity properties; and (ii) the analysis only involves the homogenized solution and its minimal regularity. We derive an interpolation error [Formula: see text] with [Formula: see text] and [Formula: see text] being the period size and the coarse mesh size, respectively, when the oversampling domain includes one layer of elements from the target coarse element.