Galerkin Finite Element Methods for Stochastic Parabolic Partial Differential Equations

We study the finite element method for stochastic parabolic partial differential equations driven by nuclear or space-time white noise in the multidimensional case. The discretization with respect to space is done by piecewise linear finite elements, and in time we apply the backward Euler method. The noise is approximated by using the generalized L2-projection operator. Optimal strong convergence error estimates in the L2 and $\dot{H}^{-1}$ norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The computational analysis and numerical example are given.

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