Scales of ε-uniform a priori estimates for nonstationary convection-dominated diffusion problems

Abstract

Convection-diffusion problems of practical interest are often convection-dominated in the sense that the diffusion parameter e satisfies 0<e≪1. Standard error estimates for numerical approximations of such problems mostly contain constants which depend either reciprocally on e or at least on higher order norms of the solution. These norms possibly increase to infinity in the hyperbolic limit e→0. For the Lagrange-Galerkin scheme such error estimates have been proven by Douglas and Russell, Pironneau or Suli. In the particular case of the Navier-Stokes equations Johnson, Rannacher and Boman have observed that some of those constants which arise in existing analyses depend exponentially on the Reynolds number. Consequently, standard error analyses have no real meaning in the case of convection-dominance. Recently, Bause and Knabner have developed a rigorous e-uniform convergence theory for finite element and Lagrange-Galerkin approximations of convection-dominated diffusion problems. The error analysis is heavily based on e-uniform a priori estimates for the solution of the continuous problems. In their work such e-uniform estimates are established in the Sobolev spacesL2(Ω) andH2(Ω) in a Lagrangian framework by transforming the convection-diffusion problem into subcharacteristic coordinates. To obtain the e-uniformH2-bound, strong conditions about the right-hand side which cannot realistically be assumed in practice are supposed. In this paper we demonstrate how the method of complex interpolation of Banach spaces can be employed to derive elegantly e-uniform a priori estimates in intermediate Sobolev spaces of fractional order. We will observe that in particularHα-regularity of the data has to be supposed to get an e-uniformHα-bound for the solution. To derive the desired estimates we use several complex interpolation results which we provide first.