A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations
For researchers in numerical analysis of stochastic PDEs, this method relaxes smoothness assumptions on nonlinearities while maintaining convergence rates.
The paper develops a novel fully discrete numerical method for semilinear stochastic evolution equations, combining a Galerkin finite element method with a randomized Runge-Kutta scheme. It achieves the same temporal convergence order as Milstein-Galerkin methods without requiring differentiability of the nonlinearity.
In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.