An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations. (8th December 2019) - Christy Roja, J.; Tamilselvan, A.; Geetha, N.
Summary: 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. Abstract : An overlapping Schwarz method is constructed for the numerical solution of a class of system of weakly coupled convection‐diffusion equations . The domain is splitted into two overlapping subdomains. We have proposed a hybrid difference scheme 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. Our designed Schwarz method produces numerical approximations that converge in the maximum norm to the exact solution. This convergence is shown to be of almost second order. The graphs plotted in the figures are convergent curves in the maximum norm at nodal points for the different values of ϵ and N . These graphs clearly indicate that the optimal error bound is of order O ( N −k + N −2 In 3 N ).