Numerical approximation of singular-degenerate parabolic stochastic PDEs

Abstract

We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation we prove the convergence of the numerical approximation towards the unique solution. Furthermore, we construct an implementable finite element scheme for the spatial discretization of the very weak formulation and provide numerical simulations to demonstrate the practicability of the proposed discretization.

Perspective: New directions in dynamical density functional theory

“…Recent work has considered stochastic DDFT (the Dean-Kawasaki equation) [177][178][179][180][181][182][183] and the McKean-Vlasov equation (a DDFT-type model) [184][185][186][187] from a mathematical perspective. Moreover, numerical methods were developed for DDFT [188][189][190][191][192][193][194][195][196] and PFC models [197][198][199][200]. DDFT was also used to test a new Brownian dynamics simulation method [201].…”