A STUDY OF NUMERICAL POLLUTION OF THE DECOUPLED ALGORITHM FOR THE CONVECTION MODEL IN SUPERPOSED FLUID AND POROUS LAYERS

For coupling problems such as the Navier-Stokes-Darcy problem in porous and fluid layers, a natural question is that, while the orders of polynomials approximating the variables are different, could the lower order elements pollute the higher ones? In this paper, the numerical pollution of a decoupled algorithm for the unsteady convection model in superposed fluid and porous layers is studied. We first prove the existence and uniqueness of the weak solution to the convection model with Beavers-Joseph-Saffman interface conditions, by using vanishing viscosity methods together with backward-Euler time discretization. Second, an efficient decoupled algorithm of the model based on the Lagrange multiplier method is proposed, and its stability is proved under some restrictions on coefficients and initial values. Then the optimal error estimates are provided revealing that the optimal convergence orders can be achieved without using equal or similar order finite elements for the fluid velocities in fluid and porous layers. This means that there is no numerical pollution between lower and higher order elements when the matched finite element pairs are used. Finally, numerical tests are implemented to verify our theoretical results.