Stochastic Partial Differential Equations with Multiplicative Noise - Numerical simulations of strong and weak approximation errors

Abstract

A finite element Galerkin spatial discretization together with a backward Euler scheme is implemented to simulate strong error rates of the homogeneous stochastic heat equation with multiplicative trace class noise in one dimension. For the noise, two different operators displaying different degrees of regularity are considered, one of which is of Nemytskii type. The discretization scheme is extended to include discretization of the covariance operator of the Q-Wiener process driving the equation. The results agree with the theory. Furthermore, for exploratory reasons, weak error rates are also simulated using the classical Monte Carlo method and the multilevel Monte Carlo method.