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 L ₂ -projection operator. Optimal strong convergence error estimates in the L ₂ 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.

In this paper, we consider the finite element method for time fractional partial differential equations. The existence and uniqueness of the solutions are proved by using the Lax-Milgram Lemma. A time stepping method is introduced based on a quadrature formula approach. The fully discrete scheme is considered by using a finite element method and optimal convergence error estimates are obtained. The numerical examples at the end of the paper show that the experimental results are consistent with our theoretical results.

We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin, Lazarov, and Zhou [IMA J. Numer. Anal., 36 (2016), pp. 197--221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has a convergence rate of $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.