Finite element and difference approximation of some linear stochastic partial differential equations.

*(English)*Zbl 0907.65147A 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 |