Optimal $$L^2, H^1$$ L 2 , H 1 and $$L^\infty $$ L ∞ analysis of finite volume methods for the stationary Navier–Stokes equations with large data

Abstract

Previous work on the stability and convergence analysis of numerical methods for the stationary Navier---Stokes equations was carried out under the uniqueness condition of the solution, which required that the data be small enough in certain norms. In this paper an optimal analysis for the finite volume methods is performed for the stationary Navier---Stokes equations, which relaxes the solution uniqueness condition and thus the data requirement. In particular, optimal order error estimates in the $$H^1$$ H 1 -norm for velocity and the $$L^2$$ L 2 -norm for pressure are obtained with large data, and a new residual technique for the stationary Navier---Stokes equations is introduced for the first time to obtain a convergence rate of optimal order in the $$L^2$$ L 2 -norm for the velocity. In addition, after proving a number of additional technical lemmas including weighted $$L^2$$ L 2 -norm estimates for regularized Green's functions associated with the Stokes problem, optimal error estimates in the $$L^\infty $$ L ? -norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor $$O(|\log h|)$$ O ( | log h | ) for the stationary Naiver---Stokes equations.