Non-Uniform Semi-Discretization of Linear Stochastic Partial Differential Equations in R

We construct and analyze approximations for linear non-degenerate parabolic stochastic partial differential equations that have solutions with values in weighted Sobolev spaces. These function spaces consist of Sobolev functions for which its derivatives up to order m and the function itself are square integrable and have at most a decay that is controlled by r. We employ spatial semi-discretizations based on finite grids for the approximation problem, where uniform as well as non-uniform smooth grids are considered. These semi-discretization schemes are based on a finite difference method, which is accelerated by the application of the Richardson extrapolation. Finally, in the last step of the construction, a suitable linear interpolation operator is applied to this semi-discrete scheme to get a global approximation. We establish upper bounds for global 2-norm errors for these schemes in terms of the parameters m and r and of the number of points of the respective grids. As a main result, suitable non-uniform smooth grids turn out to be superior to uniform grids in the following sense: For the same number of points the upper bound for the non-uniform grids is asymptotically smaller than the corresponding quantity for the uniform grids. As an important subproblem, we consider the problem of function approximation in weighted Sobolev spaces in the worst case setting. We determine the complexity of this problem and give an asymptotically optimal interpolation algorithm that is based on finite non-uniform smooth grids. If uniform grids are used instead, we will not obtain the optimal rate of convergence. These interpolations are exactly those, which are applied in the last step of the construction of the approximations for the stochastic partial differential equations.