Numerical analysis of stochastic advection-diffusion equation via Karhunen-Loéve expansion

In this work, we present a numerical analysis of a probabilistic approach to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More precisely, we consider the flow problem in a random porous medium coupled with the advection-diffusion equation and we are interested in the approximation of the mean spread and the mean dispersion of the solute. The conductivity field is represented by a Karhunen-Loeve (K-L) decomposition of its logarithm. The flow model is solved using a mixed finite element method in the physical space. The advection-diffusion equation is computed thanks to a probabilistic particular method, where the concentration of the solute is the density function of a stochastic process. This process is solution of a stochastic differential equation (SDE), which is discretized using an Euler scheme. Then, the mean of the spread and of the dispersion are expressed as functions of the approximate stochastic process. A priori error estimates are established on the mean of the spread and of the dispersion. Numerical examples show the effectiveness of the approach.