Optimal error estimates of a second-order fully decoupled finite element method for the nonstationary generalized Boussinesq model

In this paper, we develop and analyze a fully decoupled finite element method for the non- stationary generalized Boussinesq equations, where the viscosity and thermal conductivity depend on the temperature. Based on some subtle implicit-explicit treatments for the nonlinear coupling terms, we develop a second-order in time, fully decoupled, linear and unconditionally energy stable scheme for solving this system. The unconditional stability of the fully discrete scheme with finite element approximation is proved. The optimal L-2-error estimates are analyzed for temperature-dependent thermal conductivity system. Numerical experiments are presented to illustrate the convergence, accuracy and applicability of the proposed numerical scheme.