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Finite element and difference approximation of some linear stochastic partial differential equations. (English) Zbl 0907.65147
A linear elliptic differential equation with additive white noise and a linear parabolic equation with space and time white noise are considered. The solutions are defined by integral equations which contain the free function. If the noise processes are approximated by piecewise constant random processes then the solutions of the above stochastic equations have more regularity and approximate the solutions of the original problems. The approximating equations are solved by difference and finite element methods. Convergence rates are given. Numerical experiments show that the finite element methods are computationally more efficient.
Reviewer: W.Grecksch (Halle)

MSC:
65C99 Probabilistic methods, stochastic differential equations
35R60 PDEs with randomness, stochastic partial differential equations
35J25 Boundary value problems for second-order elliptic equations
35K15 Initial value problems for second-order parabolic equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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