An overlapping Schwarz method for singularly perturbed fourth-order convection-diﬀusion type
In this paper, we have constructed an iterative numerical method based on an overlapping Schwarz procedure with uniform mesh for singularly perturbed fourth-order of convection-diﬀusion type. 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 while on the non-layer region we use the mid-point diﬀerence scheme on a uniform mesh. It is shown that the method produces numerical approximations which converge in the maximum norm to the exact solution. We prove that, when appropriate subdomains are used the method produces convergence of almost second-order. Furthermore, it is shown that, two iterations are suﬃcient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results.
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