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Abstract

We design a multiscale model reduction framework within the hybridizable discontinuous Galerkin finite element method. Our approach uses local snapshot spaces and local spectral decomposition following the concept of Generalized Multiscale Finite Element Methods. We propose several multiscale finite element spaces on the coarse edges that provide a reduced dimensional approximation for numerical traces within the HDG framework. We provide a general framework for systematic construction of multiscale trace spaces. Using local snapshots, we avoid high dimensional representation of trace spaces and use some local features of the solution space in constructing a low dimensional trace space. We investigate the solvability and numerically study the performance of the proposed method on a representative number of numerical examples.

5. Conclusion

In this paper, we propose spectral multiscale finite element methods for second order elliptic equations in the framework of the hybridizable discontinuous Galerkin finite element method. We propose several finite element spaces for the numerical traces. The main idea of the proposed method is to construct a low dimensional trace space MH . In the paper, we present a general framework for defining trace spaces, which consists of three steps: (1) a partition of the coarse skeleton; (2) a construction of a local snapshot space; (3) a construction of the offline space and consequently a construction of the numerical trace space. Within this framework, we propose and test two classes of coarse spaces. The first class uses the boundaries of the coarse-grid subdomain to construct trace space, while the second class constructs the traces on the faces that are strictly within coarse blocks. In this regard, the second approach uses an oversampling technique where the information in larger domains is used in constructing multiscale basis functions. The presented numerical results for both methods show that one can achieve a good accuracy with a few degrees of freedom along each edge and the approaches that use oversampling provide better accuracy. In our future work, we plan to present detailed error analysis.