On the convergence analysis of MsFEM with oversampling: Interpolation error

In this paper, we investigate the approximation properties of two types of multiscale finite element methods with oversampling as proposed in [Hou \& Wu, {\textit{J. Comput. Phys.}}, 1997] and [Efendiev, Hou \& Wu, \textit{SIAM J. Numer. Anal.}, 2000] 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 $L^2$-accuracy. The distinct features of the analysis, in the setting of highly oscillatory periodic coefficients, include: (i) The analysis is independent of the first-order corrector or the solutions to the cell problems, and thus independent of their regularity properties; (ii) The analysis only involves the homogenized solution and its minimal regularity. We derive an interpolation error $\mathcal{O}\left(H+\frac{\epsilon}{H}\right)$ with $\epsilon$ and $H$ being the period size and the coarse mesh size, respectively, when the oversampling domain includes one layer of elements from the target coarse element.