  In this paper, we construct a combined multiscale finite element method
(MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the
multiscale problems which may have singularities in some special portions of
the computational domain. For example, in the simulation of steady flow
transporting through highly heterogeneous porous media driven by extraction
wells, the singularities lie in the near-well regions. The basic idea of the
combined method is to utilize the traditional finite element method (FEM)
directly on a fine mesh of the problematic part of the domain and using the
LOD-based MsFEM on a coarse mesh of the other part. The key point is how to
define local correctors for the basis functions of the elements near the coarse
and fine mesh interface, which require meticulous treatment. The proposed
method takes advantages of the traditional FEM and the LOD-based MsFEM, which
uses much less DOFs than the standard FEM and may be more accurate than the
LOD-based MsFEM for problems with singularities. The error analysis is carried
out for highly varying coefficients, without any assumptions on scale
separation or periodicity. {Numerical examples with periodic and random highly
oscillating coefficients}, as well as the multiscale problems on the L-shaped
domain, and multiscale problems with high-contrast channels or
well-singularities are presented to demonstrate the efficiency and accuracy of
the proposed method.
