Regularity and Derivative Bounds for a Convection-Diffusion Problem with Neumann Boundary Conditions on Characteristic Boundaries

Abstract

A convection-diffusion problem is considered on the unit square, with convection parallel to two of the square’s sides. Dirichlet conditions are imposed on the inflow and outflow boundaries, with Neumann conditions on the other two sides. No assumption is made regarding the corner compatibility of the data. The regularity of the solution is expressed precisely in terms of the regularity and compatibility of the data. Pointwise bounds on all derivatives of the solution are derived and their dependence on the data regularity, its corner compatibility, and on the small diffusion parameter is made explicit. These results extend previous bounds of Jung and Temam [Int. J. Numer. Anal. Model. 2 (2005) 367–408] and of Clavero, Gracia, Lisbona and Shishkin [Z. Angew. Math. Mech. 82 (2002) 631–647].