A modified local and parallel finite element method for the coupled Stokes-Darcy model with the Beavers-Joseph interface condition

In this paper, by combining a partition of unity with backtracking technique, a modified local and parallel finite element method (MLPFEM) is proposed and investigated for the coupled Stokes-Darcy problem with the Beavers-Joseph (BJ) interface condition. The well-posedness of the coupled Stokes-Darcy model with BJ interface condition is established when the parameter alpha(BJ) is small enough in Cao et al. (Commun. Math. Sci. 8(1), 1-25 2010). The MLPFEM is adopted based on its significant advantages that we only need to solve a series of local subproblems once the coarse approximation is derived. Compared with the global discontinuous solution by the algorithm in Du and Zuo (2017), the main features of our algorithm are as follows: (1) partition of unity functions are utilized to gather local approximations computed on a fine grid by local and parallel procedures to generate a global continuous solution; (2) a further global coarse correction, namely the backtracking technique, is considered to obtain optimal error bounds of the velocity field and the piezometric head in L-2 norm. Optimal error bounds are derived and numerical tests are carried out to support the theoretical analysis.