2025/07/19 by Li, Guanglian
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2507.14548
In this paper, we investigate the approximation properties of two types of multiscale finite element methods with oversampling as proposed in [Hou & Wu, J. Comput. Phys., 1997] and [Efendiev, Hou & Wu, 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 L2-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 O(H+\fracεH) with ε 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.