Stability and error analysis of IMEX-BDFk finite element schemes for the incompressible Navier-Stokes system

Abstract

In this paper, we propose and analyze a class of high-order numerical schemes within a fully discrete finite element framework for the incompressible Navier-Stokes equations with no-slip boundary conditions. The temporal discretization employs a kth-order (k=1,...,6) implicit-explicit backward difference formula (IMEX-BDFk), in which the nonlinear convection term is treated explicitly and the linear Stokes part implicitly, whereas the spatial discretization utilizes Taylor-Hood finite elements. We establish the stability and uniform boundedness of the numerical solution. We further establish optimal order error estimates in both space and time without any CFL-type condition, in the sense that the time step is independent of the spatial mesh size. In three dimensions, these include L2- and H1-norm error estimates for the velocity and L2-norm error estimates for the pressure, with temporal convergence rates up to sixth order for all variables. Numerical experiments are presented to demonstrate the effectiveness of the scheme and to confirm the theoretical convergence rates.