Quasi-Monte Carlo finite element approximation for singularly perturbed convection-diffusion problems with random velocity
Provides a rigorous error analysis and efficient numerical scheme for uncertainty quantification in convection-dominated problems, which is relevant for computational fluid dynamics and engineering applications.
This paper develops a numerical framework for solving singularly perturbed convection-diffusion problems with random velocity fields, combining finite elements, Karhunen-Loève expansion, and quasi-Monte Carlo methods. The method achieves a nearly linear optimal convergence rate independent of the singular perturbation parameter and integration dimension.
This paper studies the numerical approximation of a singularly perturbed convection-diffusion problem over a bounded polygonal domain in $\mathbb{R}^d$ ($d=2,3$), where the velocity field is modeled by a log-uniform random field, a setting typical in uncertainty quantification. We introduce a novel numerical framework for computing the expected value of the linear functionals of the solution. The approach combines a finite element discretization of the problem, a truncated Karhunen--Loève expansion to represent the stochastic velocity field, and a lattice-based quasi-Monte Carlo (QMC) method to estimate expectations over the parameter space. We provide a rigorous error analysis of the proposed scheme, establishing bounds on the mean squared error and demonstrating that the QMC method achieves a nearly linear optimal convergence rate, with a constant independent of the integration dimension. Furthermore, the convergence rate is shown to be independent of the singular perturbation parameter.