Mortar Finite Element Method for the Coupling of Time Dependent Navier-Stokes and Darcy Equations

The article discusses a nonlinear system that is dependent on time and coupled by incompressible fluid and porous media flow. Treating Darcy flow as dual-mixed form, we propose a variational formulation and prove the well-posedness of weak solutions. The discretization of domain is accomplished using a triangular mesh, with the lowest order Raviart-Thomas element utilized for Darcy equations and Bernardi-Raugel element used for Navier-Stokes equations. Using the mortar method, we construct the spaces from which numerical solutions are sought. Based on backward Euler method, we establish a fully discrete algorithm. At each single time level, the first-order convergence is demonstrated through the use of the Gronwall inequality. Numerical experiments are provided to illustrate the algorithm’s effectiveness in approximating solutions.