The Discontinuous Petrov–Galerkin methodology for the mixed Multiscale Finite Element Method

Abstract

We present the application of the Discontinuous Petrov–Galerkin (DPG) methodology for the mixed Multiscale Finite Element Method (MsFEM). The MsFEM upscaling technique relies on incorporating fine-scale features through special, in a sense optimized for approximability, trial functions while the DPG methodology allows for the selection of the optimal test functions to provide stability of the FEM approximation. The special trial functions are computed online by the solution of local boundary value problems. We improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton (element interfaces) only. We have verified by numerical tests that the proposed improvement of MsFEM reduced both the approximation error and computational cost. Moreover, it simplified the algorithm significantly. The key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape. The proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters, using an arbitrary well-posed functional setting with or without the DPG methodology.