A penalty method for approximation of the stationary Stokes–Darcy problem

In this work, the penalty method is studied for the mixed Stokes–Darcy problem, motivated by the penalty method applied to Stokes equation. This work first proposes the penalty Stokes–Darcy model at the continuous level. Then we prove that the solution of the penalty model converges strongly to the original solution as $O\left(ϵ\right)$ in which the penalty parameter is $ϵ\to 0$. What is more, the finite element method is used to solve the penalty model and the optimal error estimates are presented. Finally, several numerical tests are carried out to verify our theoretical results.