Abstract
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.
Highlights
Singular Perturbation Problems (SPPs) appear in many branches of applied mathematics, like fluid dynamics, quantum mechanics, turbulent interaction of waves and currents, electrodes theory, etc
The convergence of the numerical approximations generated by standard numerical methods applied to such
Tamilselvan problems depends adversely on the singular perturbation parameter. Most of these works have concentrated on second-order single differential equations ( [4] and the references therein), but for fourth-order equations only few results are reported in the literature [2, 15, 16, 17]
Summary
Singular Perturbation Problems (SPPs) appear in many branches of applied mathematics, like fluid dynamics, quantum mechanics, turbulent interaction of waves and currents, electrodes theory, etc. Numerical methods for singularly perturbed problems comprising domain decomposition and Schwarz iterative techniques have been examined by various authors, for example, in [1, 6, 7, 8, 9, 10, 18, 20]. In [20], an analysis of overlapping domain decomposition methods for singularly perturbed reaction-diffusion problems with distinct small positive parameters is presented. Motivated by the works of [2,10,15,16,17] we have examined experimentally the performance of Schwarz method to the fourth-order Singularly Perturbed Boundary Value Problems (SPBVPs) described as below.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
