A unified study on superconvergence analysis of Galerkin FEM for singularly perturbed systems of multiscale nature
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A unified study on superconvergence analysis of Galerkin FEM for singularly perturbed systems of multiscale nature Maneesh Kumar Singh1
· Gautam Singh2
· Srinivasan Natesan2
Received: 10 July 2020 / Revised: 26 August 2020 / Accepted: 31 August 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract We discuss the superconvergence analysis of the Galerkin finite element method for the singularly perturbed coupled system of both reaction–diffusion and convection– diffusion types. The superconvergence study is carried out by using linear finite element, and it is shown to be second-order (up to a logarithmic factor) uniformly convergent in the suitable discrete energy norm. We have conducted some numerical experiments for the system of reaction–diffusion and system of convection–diffusion models, which validate the theoretical results. Keywords Superconvergence · Finite element method · Singularly perturbed coupled system · Boundary layers · Shishkin mesh · Uniform convergence Mathematics Subject Classification 65L06 · 65L10 · 65L11 · 65L12 · 65L20
1 Introduction In this article, we consider singularly perturbed system of reaction–diffusion and convection–diffusion problems with different perturbation parameters. Due to distinct perturbation parameters, the exact solution exhibits overlapping boundary layers. Therefore, these problems are interesting and more difficult than the singularly perturbed system of boundary-value problems (BVPs) with a single parameter. These
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Srinivasan Natesan [email protected] Maneesh Kumar Singh [email protected] Gautam Singh [email protected]
1
Department of Computational and Data Sciences, Indian Institute of Science, Bangalore 560012, India
2
Department of Mathematics, Indian Institute of Technology, Guwahati 781039, India
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M. K. Singh et al.
types of problems arise in various fields of engineering and science, for example, the reaction–diffusion enzyme model, convection–diffusion enzyme model. More detailed applications of these problems can be found in the book of Pao [12]. These types of singularly perturbed systems have boundary layer behavior, thus classical numerical schemes on uniform meshes are not sufficient to provide uniformly convergent numerical results. One has to use some nonuniform meshes to overcome the difficulties due to uniform meshes. Various ε-uniform numerical methods, like finite difference, finite element and finite volume methods are available in the literature for singular perturbation problems. One can refer the books [10,13] and references therein, for more details. Stynes [9] considered the method of fitted difference on the piecewise-uniform Shishkin mesh to solve the reaction–diffusion system. The uniform convergence of finite element approximation of the reaction–diffusion system was analyzed by Linß [7]. In [17–19], Singh and Natesan analyzed robust numerical methods for various models of singularly perturbed system of parabolic problems of multiscale nature. Recently, Singh and Natesan
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