# Multi-grid Multi-Level Monte Carlo Method for Stokes-Darcy interface Model with Random Hydraulic Conductivity

## TL;DR

This paper introduces a novel multi-grid multi-level Monte Carlo method to efficiently solve the stochastic Stokes-Darcy interface model with random hydraulic conductivity, reducing computational costs while maintaining accuracy.

## Contribution

The paper develops a combined MGMLMC approach for the stochastic Stokes-Darcy model, integrating multi-grid and multi-level Monte Carlo techniques for improved efficiency.

## Key findings

- The MGMLMC method significantly reduces computational cost compared to traditional methods.
- Numerical examples confirm the accuracy and efficiency of the proposed approach.
- Theoretical analysis guarantees the existence and uniqueness of the solution.

## Abstract

In this article we develop a multi-grid multi-level Monte Carlo (MGMLMC) method for the stochastic Stokes-Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes-Darcy model to improve the fidelity as this model also considers the second and third porosity of the free flow. Then we prove the existence and uniqueness of the weak solution of the variational form. For the numerical solution, we adopt the Monte Carlo (MC) method and finite element method (FEM), for the discrete form in the probability space and physical space, respectively. In the traditional single-level Monte Carlo (SLMC) method, more accurate numerical approximate requires both larger number of samples in probability space and smaller mesh size in the physical space. Then the computational cost increase significantly as the mesh size becomes smaller for the more accurate numerical approximate. Therefore we adopt the multi-level Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space, because the number of samples decays fast while the mesh size decreases. We also develop a strategy to calculate the number of samples needed in MLMC method for the stochastic Stokes-Darcy model. Furthermore MLMC naturally provides the hierarchial grids and sufficient information on these grids for multi-grid (MG) method, which can in turn improve the efficiency of MLMC. In order to fully make use of the dynamical interaction between this two methods, we propose the multi-grid multi-level Monte Carlo method for more efficiently solving the stochastic model. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions.