Nitsche's method for the stationary Boussinesq system under mixed and nonlinear boundary conditions

Abstract

In this paper we analyze Nitsche's method for the stationary Boussinesq system with Navier's slip and a nonlinear boundary condition. Our analysis of the formulation establishes the robustness of a finite elements scheme in arbitrarily complex boundaries. The well-posedness of the discrete problem is established using fixed-point theorems under a standard smallness assumption on the data. We also provide optimal convergence rates for the approximation error. Furthermore, the efficiency and reliability of residual-based a posteriori error estimators are established. We validate our theory through several numerical tests.