2019/09/26 by Siu Wun Cheung, Cheung, Siu Wun, Eric T. Chung +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics
paper · pdf · doi:10.48550/arxiv.1909.12461
Numerical simulation of flow problems and wave propagation in heterogeneous\nmedia has important applications in many engineering areas. However, numerical\nsolutions on the fine grid are often prohibitively expensive, and multiscale\nmodel reduction techniques are introduced to efficiently solve for an accurate\napproximation on the coarse grid. In this paper, we propose an energy\nminimization based multiscale model reduction approach in the discontinuous\nGalerkin discretization setting. The main idea of the method is to extract the\nnon-decaying component in the high conductivity regions by identifying dominant\nmodes with small eigenvalues of local spectral problems, and define multiscale\nbasis functions in coarse oversampled regions by constraint energy minimization\nproblems. The multiscale basis functions are in general discontinuous on the\ncoarse grid and coupled by interior penalty discontinuous Galerkin formulation.\nThe minimal degree of freedom in representing high-contrast features is\nachieved through the design of local spectral problems, which provides the most\ncompressed local multiscale space. We analyze the method for solving Darcy flow\nproblem and show that the convergence is linear in coarse mesh size and\nindependent of the contrast, provided that the oversampling size is\nappropriately chosen. Numerical results are presented to show the performance\nof the method for simulation on flow problem and wave propagation in\nhigh-contrast heterogeneous media.\n