Uniform convergence analysis of finite difference approximations for advection–reaction–diffusion problem on adaptive grids

Abstract

A kind of singularly perturbed advection–reaction–diffusion problems with the exponential boundary layer are considered on an adaptive mesh. The existence, uniqueness and stability for the solution of the discrete problem are analysed with the maximum principle. The stability of the continuous problem is also considered. For the equidistribution problem composed by the difference scheme and equidistribution mesh equations, we establish a first-order ϵ-independent convergence rate for the numerical scheme defined on the equidistribution mesh and also an estimation for the accuracy of the solution computed on the final mesh generated by the adaptive algorithm. Numerical results are given to examine the validity of our theoretical analysis and the efficiency of the adaptive algorithm.

Keywords:

• advection–reaction–diffusion problem
• boundary layer
• mesh equidistribution
• convergence analysis
• error estimation

2010 AMS Subject Classifications :

• 65L11
• 65L12
• 65L20
• 65L70

Acknowledgements

This research is partially supported by Zhengzhou Institute of Surveying and Mapping Fund under Grant Y1009, the Gansu Sci-Tech Planning Fund under Grant 0804NKCA073, the National Basic Research (973) Program of China under Grant 2011CB706900, the Mathematical Tianyuan Foundation of China under Grant No. 11026064 and the Fund of Physics & Mathematics of Lanzhou University under Grant No. LZULL200904.