Analysis of an Efficient Parameter Uniform Domain Decomposition Approach for Singularly Perturbed Gierer-Meinhardt Type Nonlinear Coupled Systems of Parabolic Problems

This article presents an efficient domain decomposition algorithm of Schwarz waveform relaxation type for singularly perturbed Gierer-Meinhardt type nonlinear coupled systems of parabolic problems where the diffusion terms in each equation are multiplied by small parameters of different magnitudes. The magnitude of these small parameters leads to the sharpness and boundary layer behavior in the solution components. Our present algorithm considers a suitable decomposition of the domain and decouples the process of approximating the solution components at each time level. There are two different schemes proposed in this work. Specifically, the schemes use the backward Euler method combined with a suitable component-wise splitting for time discretization, while employing the central difference scheme for spatial discretization. The two numerical schemes differ in their splitting methods: Scheme 1 employs a Jacobi-type split, whereas Scheme 2 utilizes a Gauss-Seidel-type split. The exchange of information between neighboring subdomains is achieved through piecewise-linear interpolation. The convergence analysis of the algorithm is demonstrated using some auxiliary nonlinear systems. It is shown that the present procedure provides uniformly convergent numerical approximations to the solution having sharp spike components. Numerical experiments demonstrate that the considered algorithm with present discretization is more efficient in terms of accuracy and iteration counts than with the standard available approaches.