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In this paper, we examine the effectiveness of classic multiscale finiteelement method (MsFEM) (Hou and Wu, 1997; Hou et al., 1999) for mixedDirichlet-Neumann, Robin and hemivariational inequality boundary problems.Constructing so-called boundary correctors is a common technique in existingmethods to prove the convergence rate of MsFEM, while we think not reflects theessence of those problems. Instead, we focus on the first-order expansionstructure. Through recently developed estimations in homogenization theory, ourconvergence rate is provided with milder assumptions and in neat forms.

Convergence Rate of Multiscale Finite Element Method for Various Boundary Problems