A multiscale preconditioner for stochastic mortar mixed finite elements

Abstract

a b s t r a c tThe aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media. The governing equations are based on Darcy's law with a stochastic permeability field. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loève expansion. Multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions. A multiscale mortar flux basis is computed for a single permeability, called a training permeability, that captures the main characteristics of the porous media, and is used as a preconditioner for each stochastic realization. We prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size. Computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media.

Adjoint Based A Posteriori Analysis of Multiscale Mortar Discretizations with Multinumerics

“…In each case, we solve the subdomain problems using a sparse direct solver. We construct the forward and adjoint interface operators as in [26,49] and use GMRES to solve the interface problem to a tolerance of 1E-10.…”

A Family of Multipoint Flux Mixed Finite Element Methods for Elliptic Problems on General Grids

“…The multiblock variational framework is useful in designing optimal parallel solvers that utilize efficient interface multiscale bases as interface preconditioners and subdomain solvers such as algebraic multigrid. These approaches have also been shown to be convergent and efficient when applying stochastic methods for uncertainty analyses [15,16] and applying the MFMFE methods for multiscale modeling of nonlinear flow problems in porous media [17].…”

Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling

“…Since the paper focuses on three major issues, mortars, multiscale, and multinumerics, we consider three test examples, each emphasizing one of these issues. In each case we reduce the problem to a coarse scale interface operator and use the multiscale mortar basis method developed in [18,34] to solve the coarse scale interface equations.…”

“…Note that multiscale mortar techniques are especially appealing as the discretization can be reduced to a global problem only involving higher-order polynomials on the interface mortar mesh (see [8]). This leads to a parallel domain decomposition implementation [20] that can be enhanced by constructing a multiscale mortar basis as in [18] and applying multiscale preconditioners [34]. This approach leads to high computational efficiency.…”