On heterogeneous coupling of multiscale methods for problems with and without scale separation

Abstract

In this paper, we discuss partial differential equations with multiple scales for which scale resolution is needed in some subregions, while a separation of scale and numerical homogenization is possible in the remaining part of the computational domain. Departing from the classical coupling approach that often relies on artificial boundary conditions computed from some coarse grain simulation, we propose a coupling procedure in which virtual boundary conditions are obtained from the minimization of a coarse grain and a fine-scale model in overlapping regions where both models are valid. We discuss this method with a focus on interface control and a numerical strategy based on non-matching meshes in the overlap. A fully discrete a priori error analysis of the heterogeneous coupled multiscale method is derived, and numerical experiments that illustrate the efficiency and flexibility of the proposed strategy are presented.

An Optimization Based Coupling Method for Multiscale Problems

“…Then, we solve a saddle point problem with cost O(N ω1 + N Ω\ω1 ). We note that the cost of the optimization based method can be further reduced [6].…”