Lifting solutions of quasilinear convection-dominated problems

Abstract

In certain cases, quasilinear convection–diffusion–reaction equations range from parabolic to almost hyperbolic, depending on the ratio between convection and diffusion coefficients. From a numerical point of view, two main difficulties can arise related to the existence of layers and/or the non-smoothness of the coefficients of such equations. In this paper we study the steady-state solution of a convection-dominated problem. We present a new numerical method based on the idea of solving an associated modified problem, whose solution corresponds to a lifting of the solution of the initial problem. The method introduced here avoids an a priori knowledge of the layer(s) location and allows an efficient handling of the lack of smoothness of the coefficients. Numerical simulations that show the effectiveness of our approach are included.

Keywords:

• convection-dominated problem
• non-uniform meshes
• convergence

2000 AMS Subject Classifications :

• 35B45
• 65L12
• 65L20
• L5L50
• 655N50

Acknowledgements

The authors gratefully acknowledge the support of this work by the Centro de Matemática da Universidade de Coimbra. The authors also thank the anonymous referees for their helpful comments. This work was supported by CMUC and project PTDC/MAT/74548/2006.