Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
Provides theoretical convergence rates for numerical approximation of stochastic PDEs, relevant for researchers in computational stochastics.
The paper derives the weak convergence rate for a spatial finite element approximation of the nonlinear stochastic heat equation with multiplicative and additive noise, extending prior work on time discretization. The weak rate is essentially twice the strong rate.
We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.