Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients

TL;DR

This paper introduces an adaptive finite element method for numerically solving time-dependent advection-diffusion equations with random, discontinuous coefficients, effectively capturing heterogeneity in subsurface flow models.

Contribution

It presents a novel adaptive, pathwise discretization scheme that accounts for spatial discontinuities in random coefficients, improving stability and convergence.

Findings

Effective handling of discontinuous random coefficients

Enhanced stability and convergence of numerical solutions

Adaptive grid refinement based on coefficient discontinuities

Abstract

Subsurface flows are commonly modeled by advection-diffusion equations. Insufficient measurements or uncertain material procurement may be accounted for by random coefficients. To represent, for example, transitions in heterogeneous media, the parameters of the equation are spatially discontinuous. Specifically, a scenario with coupled advection- and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. For the numerical approximation of the solution, an adaptive, pathwise discretization scheme based on a Finite Element approach is introduced. To stabilize the numerical approximation and accelerate convergence, the discrete space-time grid is chosen with respect to the varying discontinuities in each sample of the coefficients, leading to a stochastic formulation of the Galerkin projection and the Finite Element basis.