An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations

Abstract

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

Overlapping Schwarz Method for a Singularly Perturbed Convection-Diffusion Problems with Two Distinct Parameters

“…Here, in this paper, overlapping Schwarz method is studied both analytically and numerically for a weakly coupled system of two singularly perturbed convection-diffusion equations with distinct perturbation parameters. If the perturbation parameters are equal, then the same argument is discussed in [4], this suggested method also produced almost second-order convergence.…”

“…In this section, we discuss an error analysis of the overlapping discrete Schwarz method. The authors in [4] have analysed the same numerical method without the help of auxiliary problems, whereas in this paper the analysis of numerical method based on the new idea of auxiliary problems. It helps the authors, to prove strongly the uniform convergence of the method in two steps, splitting the discretization error and the iteration error.…”

Adaptive mesh based efficient approximations for Darcy scale precipitation–dissolution models in porous media

“…Such a situation is encountered in a typical porous medium where the extremely small pore size causes the diffusion coefficient to be extremely small. These problems can also be considered as singularly perturbed problems for which the existing approaches available in References 15 and 16 extensively use the information on the location having large gradients, which is fixed for every time level; a priori, and unable to adopt the meshes if layer location varies in time. It is also observed that the moving mesh procedures in Reference 17; using the interpolations from previous time levels (as the meshes at every time are different); works only for the problems having layer locations at a fixed point in space.…”