Nonconforming generalized multiscale finite element methods

• We propose a framework for the nonconforming generalized multiscale methods with explicit steps concretely given to illustrate our methodology.
• Our approach is simple and noble in the sense that it is free of parameters which are essential in other existing discontinuous Galerkin methods.
• Oversampling ideas enhance approximation quality of our approach.
• Our method performs well to resolve highly heterogeneous media.
• We provide several numerical examples to confirm the efficiency of the proposed method.

A framework is introduced for nonconforming multiscale approach based on GMsFEM (Generalized Multiscale Finite Element Method). Snapshot spaces are constructed for each macro-scale block. The snapshot spaces can be based on either conforming or nonconforming elements. With suitable dimension reduction, offline spaces are constructed. Moment function spaces are then introduced to impose continuity among the local offline spaces, which results in nonconforming GMsFE spaces. Oversampling and randomized boundary condition strategies are considered. Steps for the nonconforming GMsFEM are given explicitly. Error estimates are derived. Numerical results are presented to support the efficiency of the proposed approach.