Constraint Energy Minimizing Generalized Multiscale Finite Element Method

Abstract

The dual continuum model serves as a powerful tool in the modeling of subsurface applications. It allows a systematic coupling of various components of the solutions. The system is of multiscale nature as it involves high heterogeneous and high contrast coefficients. To numerically compute the solutions, some types of reduced order methods are necessary. We will develop and analyze a novel multiscale method based on the recent advances in multiscale finite element methods. Our method will compute multiple local multiscale basis functions per coarse region. The idea is based on some local spectral problems, which are important to identify high contrast channels, and an energy minimization principle. Using these concepts, we show that the basis functions are localized, even in the presence of high contrast long channels and fractures. In addition, we show that the convergence of the method depends only on the coarse mesh size. Finally, we present several numerical tests to show the performance.

A two-level overlapping Schwarz method with energy-minimizing multiscale coarse basis functions

“…We note that the standard coarse problem based on a coarse mesh often fails to give a robust preconditioner [6]. In this paper, the new idea is that the multiscale finite element functions proposed in [3] are utilized to form a more robust coarse problem. In [3], constrained energy minimizing multiscale finite element functions are introduced for approximating the solution of a multiscale model problem and the approximate solutions are shown to converge with the errors linearly decreasing with respect to the coarse mesh size and independent of the contrast in the coefficient ρ(x).…”

“…In this paper, the new idea is that the multiscale finite element functions proposed in [3] are utilized to form a more robust coarse problem. In [3], constrained energy minimizing multiscale finite element functions are introduced for approximating the solution of a multiscale model problem and the approximate solutions are shown to converge with the errors linearly decreasing with respect to the coarse mesh size and independent of the contrast in the coefficient ρ(x). By using the constrained energy minimizing basis functions, we will form the coarse component of the two-level overlapping Schwarz method.…”

“…This paper is organized as follows. In Section 2, the constrained energy minimizing functions introduced in [3] are defined using two bilinear forms that are relevant to the two-level overlapping Schwarz framework. In Sections 3 and 4, the two-level overlapping Schwarz preconditioner equipped with the coarse problem from the constrained energy minimizing functions is proposed and its condition number bound is analyzed.…”

An analysis of the NLMC upscaling method for high contrast problems

“…Then proceeding analogously to [8] and employing the fact that C ratio is relatively small, we can conclude that if k = O(log( max{κ} H )), then we can obtain (3.20). Next, we consider the estimate for u − u glo 0 .…”

Computational multiscale methods for first-order wave equation using mixed CEM-GMsFEM

“…mixed formulation). For spatial discretization, we adopt the idea of CEM-GMsFEM presented in [9,10] and propose a multiscale method for heterogeneous wave propagation and construct multiscale spaces for both, the velocity and the pressure variables. In this research, we show the first-order convergence of the method using CEM-GMsFEM combined with the leapfrog scheme.…”