A sparse composite collocation finite element method for elliptic SPDEs. (English) Zbl 1234.35329

Bieri, Marcel

SIAM J. Numer. Anal. 49, No. 6, 2277-2301 (2011).

Summary: This work presents a stochastic collocation method for solving elliptic PDEs with random coefficients and forcing term which are assumed to depend on a finite number of random variables. The method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution’s statistics. The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable. A sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed, resulting in a substantial reduction of overall degrees of freedom. Like in the Monte Carlo approach, the algorithm results in solving a number of uncoupled, purely deterministic elliptic problems, which allows the integration of existing fast solvers for elliptic PDEs. Numerical examples on two-dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the “full composite” collocation finite element method and the Monte Carlo method.

MSC:

35R60	PDEs with randomness, stochastic partial differential equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
65C20	Probabilistic models, generic numerical methods in probability and statistics
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs

Keywords:

stochastic partial differential equations; stochastic collocation methods; Smolyak approximation; multilevel approximations; sparse tensor products