Finite element analysis of convection dominated reaction–diffusion problems Original Research Article

The numerical analysis of the CAU (Consistent Approximate Upwind) Petrov–Galerkin method of convection dominated reaction–diffusion problems is presented. The main issue in this analysis is that it considers elements with high interpolation orders and yields new definitions for the upwind functions and the local Peclet number in terms of the characteristic element h and the element interpolation order p. For regular solutions, the CAU method gets quasi-optimal convergence rate for the streamline derivative, keeping the same SUPG (Streamline Upwind Petrov–Galerkin) convergence rates. This improves the well-known h-version error analysis in [Comput. Methods Appl. Mech. Engrg. 45 (1984) 285] and the hp-version in [SIAM J. Numer. Anal. 37 (2000) 1618] for SUPG-like methods. It also improves the a priori analysis for shock-capturing methods presented in [Comput. Methods Appl. Mech. Engrg. 191 (2002) 2997].