An overlapping Schwarz method for singularly perturbed fourth-order convection-diffusion type

• Published in: Mathematical modelling and analysis Vol. 25; no. 4; pp. 661 - 679
• Main Authors: Roja, J. Christy, Tamilselvan, Ayyadurai
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 13.10.2020
• Subjects: Applied mathematics; Approximation; Boundary conditions; Boundary layers; Convection; convection-diffusion equations; Convergence; Decomposition; Equations (Mathematics); Exact solutions; Fluid dynamics; hybrid difference scheme; Iteration (Mathematics); Iterative methods; Mathematical research; Methods; Numerical analysis; Numerical methods; Perturbation theory; Schwarz method; Singular perturbation methods; singularly perturbed problems; India; convection-diffusion equations; singularly perturbed problems; Schwarz method; hybrid difference scheme
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2020.10517

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.