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Finite element methods for semilinear elliptic stochastic partial differential equations. (English) Zbl 1121.65008

The authors study finite element methods for the boundary value problem for semilinear stochastic elliptic partial differential equations driven by additive white noise \[ \begin{aligned} -\Delta u(x)+ f(u(x)) &= g(x)+\dot W(x),\quad x\in\Omega,\\ u(x) &= 0,\quad x\in\partial\Omega,\end{aligned} \] where \(\Omega\) is a bounded open set of \(\mathbb{R}^2\), \(\dot W(x)\) is a white noise.
Error estimates are established. Numerical examples are presented.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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