Optimal Error Estimates of A Decoupled Scheme Based on Two-Grid Finite Element for Mixed Stokes-Darcy Model

TL;DR

This paper provides a rigorous theoretical analysis that confirms the optimal error estimates of a two-grid finite element decoupled scheme for the mixed Stokes-Darcy model, aligning with numerical results.

Contribution

It introduces an $H^1$-orthogonal decomposition to establish the first optimal error estimates for velocity and pressure in the fluid region.

Findings

Achieved optimal error estimates for velocity and pressure.

Bridged the gap between numerical results and theoretical analysis.

Validated the effectiveness of the two-grid scheme for mixed Stokes-Darcy models.

Abstract

Although the numerical results suggest the optimal convergence order of the two-grid finite element decoupled scheme for mixed Stokes-Darcy model with Beaver-Joseph-Saffman interface condition in literatures, the numerical analysis only get the optimal error order for porous media flow and a non-optimal error order that is half order lower than the optimal one in fluid flow. The purpose of this paper is to fill in the gap between the numerical results and the theoretical analysis. By introducing an $H^1-$ orthogonal decomposition of a specific vector valued space, we obtain the optimal error estimates of the velocity and pressure in fluid flow region.