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

TL;DR

This paper introduces two innovative ensemble domain decomposition methods combined with Monte Carlo and multi-level Monte Carlo techniques to efficiently solve stochastic Stokes-Darcy coupled problems with uncertain parameters, achieving optimal convergence and computational efficiency.

Contribution

The paper develops novel ensemble domain decomposition algorithms integrated with Monte Carlo methods for fast, parallel solutions of stochastic coupled flow problems, with proven convergence and reduced computational cost.

Findings

Algorithms achieve mesh-dependent and mesh-independent convergence rates.

Multi-level Monte Carlo significantly reduces computational cost.

Numerical experiments confirm the efficiency and effectiveness of the proposed methods.

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.