In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates. K E Y W O R D Sconvection-diffusion equations, hybrid difference scheme, Schwarz method, singularly perturbed problems, weakly coupled system

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In this paper, a parameter uniform numerical method based on Shishkin mesh is suggested to solve singularly perturbed third order differential equations with a turning point exhibiting boundary layers. An error estimate is derived by using supremum norm. Numerical results are provided to illustrate the theoretical results.

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