Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

Abstract

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.

Numerical upscaling of the free boundary dam problem in multiscale high-contrast media

“…We then maintain the duality method explained before but using the approximation R 0 p (n+1) 0 ≈ p (n+1) that makes the computation more efficient since, instead of solving the full resolution linear system (13), we solve the small coarse problem (16). We mention that due to the high-contrast multiscale structure of the coefficient, we need to solve the coarse problem at the right resolution in order to obtain good approximation; see [15].…”

“…Indeed, in some cases, robust approximation properties which are independent of the contrast are required. For instance, see [16,21,22] where it is demonstrated that classical numerical upscaling methods ( [19]) do not render robust approximation properties in terms of the contrast and multiscale variations (when no scale separation is considered). Furthermore, it is shown that one basis functions per coarse node (with the usual support) is not enough to construct adequate coarse spaces [22,26].…”

“…The GMsFEMs methodology aims to construct coarse spaces for Multiscale Finite Element Methods (MsFEMs) that result in accurate coarse-scale solutions for the case of high-contrast multiscale problems and, in general, for problems with a dependence on a physical parameter that negatively affects the performance of classical numerical methods. This methodology was first developed in [15,18] based on some previous works [14,16,[20][21][22].…”

A Review of Recent Advances in Discretization Methods,a PosterioriError Analysis, and Adaptive Algorithms for Numerical Modeling in Geosciences

“…The issue of devising good preconditioners for highly heterogeneous problems is an active field of research. For a discussion on this topic as well as for an up-to-date bibliographic section [13][14][15]. When it comes to dG methods, one of the very few contributions available is the work of Ayuso de Dios and Zikatanov [16].…”

“…Starting from (15), an alternative way of recovering a discrete Poincare´inequality is to introduce a least-square penalization of interface jumps inspired by the work of Arnold [59] leading to cell-centered Galerkin (ccG) methods. For all F e F i h with F & @T 1 \ @T 2 , we introduce the jump and (weighted) average operators defined by:…”