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

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.