SDFEM for singularly perturbed boundary-value problems with two parameters
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SDFEM for singularly perturbed boundary-value problems with two parameters D. Avijit1
· S. Natesan1
Received: 11 March 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this article, we study the convergence of the streamline-diffusion finite element method (SDFEM) for singularly perturbed boundary-value problem with two parameters. We prove that the SDFEM is uniformly convergent in the discrete SD-norm, of second-order on the Shishkin mesh and almost second-order on the Duran–Shishkin mesh. Numerical results are presented to support the theoretical error estimates. Keywords Singularly perturbed two-parameter BVPs · Boundary layers · Streamline-diffusion finite element method · Shishkin mesh · Graded mesh · Uniform convergence Mathematics Subject Classification 34B08 · 34E15 · 65L10 · 65L11 · 65L20 · 65L50 · 65L60 · 65L70
1 Introduction Singularly perturbation problems (SPPs) arise in various fields of engineering and science. For example, in mathematical modeling of steady and unsteady viscous flow problems with high Reynolds number, the convective heat transport problems with high Péclet numbers, linearized Burger’s equation or Navier–Stokes equations at high Reynolds number and the drift diffusion equation of semiconductor device modeling, etc. The solution of SPPs exhibits boundary layers, where the solution has steep gradients, and the solution varies rapidly in the boundary layers, and behaves smoothly outside the boundary layers.Therefore, classical numerical methods, like finite difference/element/volume methods fail to yield uniformly convergent numerical solutions
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S. Natesan [email protected] D. Avijit [email protected]
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Department of Mathematics, Indian Institute of Technology, Guwahati 781039, India
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D. Avijit, S. Natesan
on uniform meshes, and one requires to use layer-adapted nonuniform meshes to obtain satisfactory numerical approximate solutions. There are several uniformly convergent numerical methods including finite difference, finite element, finite volume methods are available in the literature to solve singularly perturbed ODEs and PDEs, for more details one can refer the books by [7,17]. Natesan and Ramanujam [14] proposed an initial-value technique to solve singularly perturbed turning point problems. In [21], the authors devised a robust numerical method for two-parameter singularly perturbed two-point boundary-value problems (BVPs) with a discontinuous source term. In [6], Das and Natesan proposed a uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion BVPs with Robin-type boundary conditions. To obtain the approximate serious solutions, Turkyilmazoglu applied the well-known homotopy method for nonlinear singularly perturbed BVPs in [25] and for parameterized unperturbed and singularly perturbed two-point BVPs in [26]. Two-parameter SPPs arise in chemical reactor theory, lubrication theories, DCmotor analysis, hydrological system analysis, and so on [15]. The solution of these problems ex
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