Title:          A multiscale preconditioner for stochastic mortar mixed finite elements


Authors:        M. F. Wheeler, T. Wildey, and I. Yotov


Source:         University of Pittsburgh Tech Report TR-MATH 09-


Abstract: The aim of this paper is to introduce a new approach to efficiently
perform uncertainty quantification for flow in porous media through
stochastic modeling.  The governing equations are based on Darcy's law
with stochastic permeability. Starting from a specified covariance
relationship, the log permeability is decomposed using a truncated
Karhunen-Loeve 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 basis is computed for a single permeability that captures the
main characteristics of the porous media, called a training
permeability, and 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.