Stochastic partial differential equations (SPDEs) play fundamental and important roles in many problems and their numerical simulations become useful and important in practical applications. The mathematical treatment of SPDEs represents however a greater challenge in comparison to deterministic partial differential equations.
We propose in this talk an overview of finite element approximations of SPDEs, particularly those singular SPDEs modeling fluid flows such as the stochastic pressure equations or the stochastic shallow water equations that involve additive and multiplicative white noises using the Wick product. We give existence and uniqueness results for the continuous linear problem and its approximation by finite element method. Optimal error estimates are derived and algorithmic aspects of the method are discussed. Our method will reduce the problem of solving SPDEs by solving a set of deterministic ones. Moreover, one can reconstruct particular realizations of the solution directly from Wiener chaos expansions once the coefficients are available. Finally, we present some numerical test problems.
Hassan Manouzi received his Ph.D. degree from Laval University, after graduate studies (master degree and "doctorat troisieme cycle") in numerical analysis from Université Paris 6. He currently holds the position of full professor in applied mathematics in the department of Mathematics and Statistic at Laval University, Quebec, Canada. His research is focused on applied mathematics, numerical analysis and scientific computing. He also teaches undergraduate and graduate courses in applied mathematics.
His significant work relates to the design of efficient numerical methods for the solution of both deterministic and stochastic partial differential equations (SPDEs) that arise in the description of natural and industrial processes. The focus of his recent work has been on the development of novel numerical methods for SPDEs driven by white noises.
A major component of his work is on the analysis and finite element approximations of stochastic pressure equation for the porous media, stochastic Navier-Stokes equations driven by additive and multiplicative white noises, stochastic shallow water equations with uncertain bathymetry and of stochastic optimal control problems constrained by stochastic linear elliptic PDEs. More recently, he investigated numerical methods for uncertainty quantification in the field of optical photonics.