TY - JOUR

T1 - Solving log-transformed random diffusion problems by stochastic galerkin mixed finite element methods

AU - Ullmann, Elisabeth

AU - Powell, Catherine E.

N1 - Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics and American Statistical Association.

PY - 2015

Y1 - 2015

N2 - Stochastic Galerkin finite element discretizations of PDEs with stochastically nonlinear coefficients lead to linear systems of equations with block dense matrices. In contrast, stochastic Galerkin finite element discretizations of PDEs with stochastically linear coefficients lead to linear systems of equations with block sparse matrices, which are cheaper to manipulate and precondition in the framework of Krylov subspace iteration. In this paper we focus on mixed formulations of second-order elliptic problems, where the diffusion coefficient is the exponential of a random field and the priority is to approximate the flux. We build on the previous work [E. Ullmann, H. C. Elman, and O. G. Ernst, SIAM J. Sci. Comput., 34 (2012), pp. A659-A682] and reformulate the PDE model as a first-order system in which the logarithm of the diffusion coefficient appears on the left-hand side. We apply a stochastic Galerkin mixed finite element method and discuss block triangular and block diagonal preconditioners for use with GMRES iteration. In particular, we analyze a practical approximation to the Schur complement of the Galerkin matrix and provide spectral inclusion bounds. Numerical experiments reveal that the preconditioners are completely insensitive to the spatial mesh size and are only slightly sensitive to the statistical parameters of the diffusion coefficient. As a result, the computational cost of approximating the flux when the diffusion coefficient is stochastically nonlinear grows only linearly with respect to the total problem size.

AB - Stochastic Galerkin finite element discretizations of PDEs with stochastically nonlinear coefficients lead to linear systems of equations with block dense matrices. In contrast, stochastic Galerkin finite element discretizations of PDEs with stochastically linear coefficients lead to linear systems of equations with block sparse matrices, which are cheaper to manipulate and precondition in the framework of Krylov subspace iteration. In this paper we focus on mixed formulations of second-order elliptic problems, where the diffusion coefficient is the exponential of a random field and the priority is to approximate the flux. We build on the previous work [E. Ullmann, H. C. Elman, and O. G. Ernst, SIAM J. Sci. Comput., 34 (2012), pp. A659-A682] and reformulate the PDE model as a first-order system in which the logarithm of the diffusion coefficient appears on the left-hand side. We apply a stochastic Galerkin mixed finite element method and discuss block triangular and block diagonal preconditioners for use with GMRES iteration. In particular, we analyze a practical approximation to the Schur complement of the Galerkin matrix and provide spectral inclusion bounds. Numerical experiments reveal that the preconditioners are completely insensitive to the spatial mesh size and are only slightly sensitive to the statistical parameters of the diffusion coefficient. As a result, the computational cost of approximating the flux when the diffusion coefficient is stochastically nonlinear grows only linearly with respect to the total problem size.

KW - Convection-diffusion

KW - Generalized saddle point problems

KW - Mixed finite elements

KW - PDEs with random data

KW - Preconditioning

KW - Schur complement approximation

KW - Stochastic finite elements

UR - http://www.scopus.com/inward/record.url?scp=85018000181&partnerID=8YFLogxK

U2 - 10.1137/14100097X

DO - 10.1137/14100097X

M3 - Article

AN - SCOPUS:85018000181

SN - 2166-2525

VL - 3

SP - 509

EP - 534

JO - SIAM-ASA Journal on Uncertainty Quantification

JF - SIAM-ASA Journal on Uncertainty Quantification

IS - 1

ER -