Numerical analysis of random derivative equations, applications to hydrogeology.

Summary This work presents some results concerning deterministic and probabilistic numerical methods for partial differential equations with random coefficients, with applications to hydrogeology. We first focus on the flow equation in a porous medium in steady state with a homogeneous lognormal permeability coefficient, including the case of a weakly regular covariance function. We establish estimates in the strong and weak sense of the error committed on the solution by truncating the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates from which we derive an extension of the existing error estimate for the stochastic collocation method, as well as an error estimate for a multilevel Monte-Carlo method. Finally, we focus on the coupling of the flow equation considered above with an advection-diffusion equation, in the case of large uncertainties and a small correlation length. A numerical analysis of a numerical method for calculating the average velocity at which the area contaminated by a pollutant expands is proposed. It is a Monte-Carlo method combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation.