Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations

Abstract

In this paper, the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme. The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally, while the previous works require certain time-step restrictions. The analysis is based on an iterated time-discrete system, with which the error function is split into a temporal error and a spatial error. The $\tau$-independent ($\tau$ is the time stepsize) error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis, which implies that the numerical solution in $L^{\infty}$-norm is bounded. Thus optimal error estimates can be obtained in a traditional way. Numerical results are provided to confirm the theoretical analysis.