An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions
Li‐Bin Liu, Ying Liang, Xiaobing Bao, Honglin Fang
Abstract
Abstract A system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:math> . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.