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.
Keywords: Flow in heterogeneous media, fractured media, porous media, jump-diffusion coefficient, non-continuous random fields, parabolic equation, finite element method, uncertainty quantification
@article{M2AN_2022__56_5_1545_0,
author = {Barth, Andrea and Stein, Andreas},
title = {Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1545--1578},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {5},
doi = {10.1051/m2an/2022054},
mrnumber = {4454163},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022054/}
}
TY - JOUR AU - Barth, Andrea AU - Stein, Andreas TI - Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1545 EP - 1578 VL - 56 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022054/ DO - 10.1051/m2an/2022054 LA - en ID - M2AN_2022__56_5_1545_0 ER -
%0 Journal Article %A Barth, Andrea %A Stein, Andreas %T Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1545-1578 %V 56 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022054/ %R 10.1051/m2an/2022054 %G en %F M2AN_2022__56_5_1545_0
Barth, Andrea; Stein, Andreas. Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 5, pp. 1545-1578. doi: 10.1051/m2an/2022054
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