Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems

Abstract

By introducing the term $-{\varepsilon} ^2 \Delta p$ into the equation of continuity and an additional Neumann boundary condition for the pressure $p$, a strongly elliptic system is obtained which is a singular perturbation of the Stokes system. We use parameter-dependent Sobolev norms to derive asymptotically precise estimates for solutions to the perturbed problem as ${\varepsilon} \searrow 0$. This results in optimal estimates for the difference between solutions to both problems; such estimates are not available by the usually applied energy methods. Under additional regularity assumptions for the data, for the energy estimates, the order of convergence with respect to ${\varepsilon}$ is improved, and convergence in $H^{s+1}$ and $H^s$ norms is obtained for the velocity and pressure with $s\in [0,3/2)$. We verify the asymptotic precision of the estimates by constructing the boundary layers.