Local And Parallel Finite Element Methods For The Coupled Stokes-Darcy Model With The Beavers-Joseph Interface Condition

In this paper,a modified local and parallel finite element method（MLPFEM）for the coupled Stokes-Darcy model with Beavers-Joseph（BJ）interface condition is proposed and studied by combining the partition of unity method and backtracking technique.This method is a successful combination of the two-grid method and the domain decomposition method.It is consistent with the core idea of the two-grid method,that is,the numerical solution is divided into two parts,the low frequency component and the high frequency component,which are obtained on the coarse grid and the fine grid respectively.The local and parallel finite element method decouples the coupled problem into a series of sub-problems for parallel solving,thus improving the computational efficiency.Most studies focus on the coupled Stokes-Darcy model with Beavers-Joseph-Saffman（BJS）interface condition.However,the BJS interface condition ignores the fluid velocity in the porous medium region on the interface,and the error between the model and the actual problem is higher than that of the model with the BJ interface condition.It was not until the reference[3]confirmed that the coupled Stokes-Darcy model with BJ interface condition is well-posed when theis sufficiently small.Based on the significant advantages of MLPFEM,once the approximate solution on the coarse grid is obtained,we only need to solve a series of local subproblems in parallel.Compared with the algorithm in[13],the main features of this algorithm are:（1）using the partition of unity function,calculate the local residuals on a fine grid by the local and parallel algorithm,and finally get the global continuous solution;（2）considering the global coarse mesh correction,namely backtracking technique,the optimal error bounds of velocity and piezometric head in²norm are obtained.The optimal error rate is given and demonstrated,and numerical experiments are carried out to support the theoretical analysis.In addition,a local and parallel finite element method for the non-stationary Stokes-Darcy model with BJ interface condition is considered.The space is dis-cretized by finite element method to get the semi-discrete scheme,and the prior error estimation and the error convergence order of the semi-discrete solution are given.The time is discretized by backward Euler method,the fully discrete scheme is obtained,and the error analysis of the final numerical solution is deduced.