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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.
MSC:
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)
35R60PDEs with randomness, stochastic PDE
35J25Second order elliptic equations, boundary value problems
35K15Second order parabolic equations, initial value problems
60H15Stochastic partial differential equations
65N06Finite difference methods (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)