Volume 56, Number 5, September-October 2022
|Page(s)||1545 - 1578|
|Published online||20 July 2022|
Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients
Institute of Applied Analysis and Numerical Simulation/SimTech, University of Stuttgart, Stuttgart, Germany
2 Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland
* Corresponding author: email@example.com
Accepted: 6 June 2022
As an extension to the well-established stationary elliptic partial differential equation (PDE) with random continuous coefficients we study a time-dependent advection-diffusion problem, where the coefficients may have random spatial discontinuities. In a subsurface flow model, the randomness in a parabolic equation may account for insufficient measurements or uncertain material procurement, while the discontinuities could represent transitions in heterogeneous media. Specifically, a scenario with coupled advection and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. The respective coefficient functions allow a very flexible modeling, however, they also complicate the analysis and numerical approximation of the corresponding random parabolic PDE. We show that the model problem is indeed well-posed under mild assumptions and show measurability of the pathwise solution. For the numerical approximation we employ a sample-adapted, pathwise discretization scheme based on a finite element approach. This semi-discrete method accounts for the discontinuities in each sample, but leads to stochastic, finite-dimensional approximation spaces. We ensure measurability of the semi-discrete solution, which in turn enables us to derive moments bounds on the mean-squared approximation error. By coupling this semi-discrete approach with suitable coefficient approximation and a stable time stepping, we obtain a fully discrete algorithm to solve the random parabolic PDE. We provide an overall error bound for this scheme and illustrate our results with several numerical experiments.
Mathematics Subject Classification: 65M60 / 60H25 / 60H30 / 60H35 / 35R60 / 58J65 / 35K10
Key words: Flow in heterogeneous media / fractured media / porous media / jump-diffusion coefficient / non-continuous random fields / parabolic equation / finite element method / uncertainty quantification
© The authors. Published by EDP Sciences, SMAI 2022
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