2021/05/09 by Yifan Chen, Thomas Y. Hou, Chen, Yifan +3 · 1 citation
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #Advanced Numerical Methods in Computational Mathematics
paper · pdf · doi:10.48550/arxiv.2105.04080
In this paper, we present a multiscale framework for solving the Helmholtz\nequation in heterogeneous media without scale separation and in the high\nfrequency regime where the wavenumber k can be large. The main innovation is\nthat our methods achieve a nearly exponential rate of convergence with respect\nto the computational degrees of freedom, using a coarse grid of mesh size\nO(1/k) without suffering from the well-known pollution effect. The key idea\nis a non-overlapped domain decomposition and its associated coarse-fine scale\ndecomposition of the solution space that adapts to the media property and\nwavenumber; this decomposition is inspired by the multiscale finite element\nmethod (MsFEM). We show that the coarse part is of \low complexity in\nthe sense that it can be approximated with a nearly exponential rate of\nconvergence via local basis functions, due to the compactness of a restriction\noperator that maps Helmholtz-harmonic functions to their interpolation residues\non edges, while the fine part is \local such that it can be computed\nefficiently using the local information of the right hand side. The combination\nof the two parts yields the overall nearly exponential rate of convergence of\nour multiscale method. Our method draws many connections to multiscale methods\nin the literature, which we will comment in detail. We demonstrate the\neffectiveness of our methods theoretically and numerically; an exponential rate\nof convergence is consistently observed and confirmed. In addition, we observe\nthe robustness of our methods regarding the high contrast in the media\nnumerically. We specifically focus on 2D problems in our exposition since the\ngeometry of non-overlapped domain decomposition is simplest to explain in such\ncases; generalizations to 3D will be outlined at the end.\n