New Ensemble Domain Decomposition Method for the Steady-state Random Stokes-Darcy Coupled Problems with Uncertain Parameters

TL;DR

This work tackles efficient computation for steady-state Stokes-Darcy coupling with uncertain hydraulic conductivity by marrying ensemble domain decomposition with Monte Carlo sampling. The authors develop two algorithms: MC ensemble DDM, which handles stochastic inputs via MC sampling, and MLMC ensemble DDM, which reduces sampling cost in probability space through a multi-level approach, while sharing a common coefficient matrix across samples to exploit parallelism. They derive convergence results under Robin parameter choices, identify mesh-dependent vs mesh-independent regimes, and provide optimal Robin parameters to accelerate convergence. Finite element discretizations are analyzed, showing mesh-dependent convergence when $\\gamma_f=\\gamma_p$ and mesh-independent convergence when $\\gamma_f<\\gamma_p$. Numerical experiments in 2D/3D demonstrate accuracy, efficiency gains, and clear advantages of MLMC over standard MC, confirming the practical impact of the proposed methods for uncertain Stokes-Darcy simulations.

Abstract

This paper presents two novel ensemble domain decomposition methods for fast-solving the Stokes-Darcy coupled models with random hydraulic conductivity and body force. To address such random systems, we employ the Monte Carlo (MC) method to generate a set of independent and identically distributed deterministic model samples. To facilitate the fast calculation of these samples, we adroitly integrate the ensemble idea with the domain decomposition method (DDM). This approach not only allows multiple linear problems to share a standard coefficient matrix but also enables easy-to-use and convenient parallel computing. By selecting appropriate Robin parameters, we rigorously prove that the proposed algorithm has mesh-dependent and mesh-independent convergence rates. For cases that require mesh-independent convergence, we additionally provide optimized Robin parameters to achieve optimal convergence rates. We further adopt the multi-level Monte Carlo (MLMC) method to significantly lower the computational cost in the probability space, as the number of samples drops quickly when the mesh becomes finer. Building on our findings, we propose two novel algorithms: MC ensemble DDM and MLMC ensemble DDM, specifically for random models. Furthermore, we strictly give the optimal convergence order for both algorithms. Finally, we present several sets of numerical experiments to showcase the efficiency of our algorithm.

Figures (7)

• Figure 2.1: The global domain $\overline{\Omega}$ consisting of the fluid domain $\Omega_f$ and the porous media region $\Omega_p$ separated by the interface $\Gamma$.
• Figure 7.1: The iterates of the Ensemble DDM (J=3) with different Robin parameters $\gamma_{f}$ and $\gamma_{p}$ while $h=\frac{1}{32}$.
• Figure 7.2: The iterates of the Ensemble DDM (J=3) with different Robin parameters $\gamma_{f}$ and $\gamma_{p}$ while $h=\frac{1}{32}$.
• Figure 7.3: The Mento Calo convergence result of the Stokes (left) and Darcy (right) with $\gamma_{f}^{\ast}$ and $\gamma_{p}^{\ast}$.
• Figure 7.4: The velocity streamlines of velocity expectations of the MC ensemble DDM (left), the traditional DDM (middle) and the MLMC ensemble DDM (right).

Theorems & Definitions (13)

• Lemma 3.1
• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• Theorem 4.4
• proof
• Theorem 5.1
• proof