Orateur
Description
We prove that a finite volume approximation of a stochastic heat equation on a polygonal two-dimensional domain $\Lambda$, with multiplicative transport noise in the Stratonovich sense, satisfies a pathwise discrete energy estimate. Coupled with two possible strategies for the time discretization, this yields, almost surely, an $L^\infty((0,T);L^2(\Lambda))$ bound on the discrete solution $u$, as well as an $L^2((0,T);L^2(\Lambda))$ bound on $\nabla u$.
These estimates are expected to be sufficient to pass to the limit in the numerical scheme and to prove convergence of the approximation toward the unique solution of the limiting equation. This is not a completely standard result since it is expected that there is a residual term in the limit equation which has the form of an additional second-order viscous term.
Finally, I will illustrate the performance of the finite volume method with numerical simulations based on a two-point flux approximation.
This is a joint work with Anne De Bouard and Ludovic Goudenège.