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 -