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A homogenization approach is one of effective strategies to solve multiscaleelliptic problems approximately. The finite element heterogeneous multiscalemethod (FEHMM) which is based on the finite element makes possible to simulatesuch process numerically. In this paper we introduce a FEHMM scheme formultiscale elliptic problems based on nonconforming spaces. In particular weuse the noconforming element with the periodic boundary condition introduced inthe companion paper. Theoretical analysis derives a priori error estimates inthe standard Sobolev norms. Several numerical results which confirm ouranalysis are provided.

$P_1$--Nonconforming Quadrilateral Finite Element Space with Periodic  Boundary Conditions: Part II. Application to the Nonconforming Heterogeneous  Multiscale Method