Error analysis of a fully discrete finite element variational multiscale method for the natural convection problem

For the natural convection problem, we propose a new projection-based finite element variational multiscale method by defining the stabilization terms via two local Gauss integrations at the element level. Based on the implicit backward Euler and implicit Crank–Nicolson schemes for temporal discretization and stabilized mixed finite element spatial discretization, we establish two numerical schemes for the natural convection problem. Unconditional stabilities of the two numerical schemes are proved. We derive error bounds of the fully discrete solution which are first and second order in time, respectively. The optimal error estimates in space could be achieved for velocity and temperature in the H1H1 semi-norm, and for pressure in the L2L2 norm with the proper choosing of stabilized parameters. However, the error estimates in space are suboptimal for velocity and temperature in the L2L2 norm. The derived theoretical results are supported by two numerical examples.