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Galerkin finite element approximations of stochastic elliptic partial differential equations. (English) Zbl 1080.65003
Two numerical methods for a linear elliptic problem \[ - \nabla ( a \nabla u) = f \] with random coefficients \(a\), \(f\) and homogeneous Dirichlet boundary conditions \(u = 0\) on \(\partial D\) are analyzed in view of the computations to approximate statistical moments of its solution \(u\). In particular, they give a priori error estimates for the computation of expected values of the solution \(u\). The first method, also called Monte Carlo-type Galerkin finite element method, uses a standard Galerkin finite element variational formulation. A Monte Carlo method exploits these approximations to compute corresponding sample averages. A second method, also called stochastic Galerkin finite element method, is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the \(h\)- or \(p\)-version, approximates the corresponding deterministic solution. The authors present a priori error estimates for both methods. They also include a comparison of the computational work required by each numerical approximation to achieve a given accuracy.
The presented a priori error estimates are useful to characterize the convergence, and also provide information to compare computational complexities of numerical methods. Such a comparison is indeed provided in the last section of the paper for the proposed Monte Carlo-type Galerkin finite element method and the second stochastic Galerkin finite element method. They conclude that if the noise is described by a small number of random parameters or if the accuracy requirement is sufficiently strict, then the stochastic Galerkin finite element method is preferred; otherwise, the Monte Carlo-type Galerkin finite element method still seems to be better. At least, intuitive conditions for an optimal selection of the numerical approximation are suggested.

65C30 Numerical solutions to stochastic differential and integral equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35R60 PDEs with randomness, stochastic partial differential equations
37H10 Generation, random and stochastic difference and differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
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