A posteriori error estimation for the Stokes–Darcy coupled problem on anisotropic discretization

Abstract

This paper presents an a posteriori error analysis for the stationary Stokes–Darcy coupled problem approximated by finite element methods on anisotropic meshes in R^N , N = 2 or 3. Korn’s inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to non-conforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for non-conforming case. With the obtained finite element solu- tions, the error estimators are constructed and based on the residual of model equations plus the stabilization terms. The lower error bound is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consid- eration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media.