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In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of a type of convection-diffusionreaction problems in two dimensions. Our finite point method has been tailored based on the local exponential basis functions. Furthermore, our TFPM satisfies the discrete maximum principle automatically. We also study the error estimates of our TFPM. We prove that our TFPM can achieve good accuracy even when the mesh size h ≫ ε for some cases without any prior knowledge of the boundary layers. Our numerical examples show the efficiency and reliability of our method. © 2012 American Mathematical Society.

Tailored finite point method based on exponential bases for convection-diffusion-reaction equation