@Article{IJNAM-22-459,
author = {Aakansha , Kumar , Sunil and Das , Pratibhamoy},
title = {Analysis of an Efficient Parameter Uniform Domain Decomposition Approach for Singularly Perturbed Gierer-Meinhardt Type Nonlinear Coupled Systems of Parabolic Problems },
journal = {International Journal of Numerical Analysis and Modeling},
year = {2025},
volume = {22},
number = {4},
pages = {459--482},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2617-8710},
doi = {https://doi.org/10.4208/ijnam2025-1020},
url = {http://global-sci.org/intro/article_detail/ijnam/24036.html}
}