Finite Element Analysis for Mass-Lumped Three-Step Taylor Galerkin Method for Time Dependent Singularly Perturbed Problems with Exponentially Fitted Splines

Sangwan, Vivek; Kumar, B. V. Rathish

doi:10.1080/01630563.2012.658484

Cited by 7 publications

(2 citation statements)

References 15 publications

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“…First, the stability and uniform convergence results for a time dependent singularly perturbed parabolic retarded differential-difference partial differential equation using 3TGFEM are discussed. This extends the work on 3TGFEM for SPDEs [29]. Next, the convergence of 3TGFEM embedded MSIM for SPPRDDE is proved.…”

Section: Introductionmentioning

confidence: 88%

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Kumar

2015

Numerical Functional Analysis and Optimization

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Monotone Schwarz iterative methods for parabolic partial differential equations are well known for their advantage of eliminating the search for an initial solution. In this article, we propose a monotone Schwarz iterative method for singularly perturbed parabolic retarded differential-difference equations based on a three-step Taylor Galerkin finite element scheme. The stability and -uniform convergence of the three-step Taylor Galerkin finite element method have been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed monotone Schwarz iterative method has been established.

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Section: Introductionmentioning

confidence: 88%

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Kumar

2015

Numerical Functional Analysis and Optimization

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“…Stynes and Riordan studied the singularly perturbed parabolic IBVP by applying the exponentially fitted difference scheme in . For the convergence of three‐step Taylor Galerkin finite element method, Sangwan and Kumar used exponentially fitted splines for the solution of singularly perturbed time‐dependent convection‐diffusion problems. To approximate the solutions of the parabolic convection‐diffusion model problems Yüzbaş and Şahin presented a numerical scheme based on Bessel's collocation method.…”

Section: Problem Statement: Preliminariesmentioning

confidence: 99%

A parameter‐uniform scheme for singularly perturbed partial differential equations with a time lag

Kumar

Kumari

2019

Numerical Methods Partial

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A numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer the extended cubic B-spline basis functions consisting of a free parameter are used on a fitted-mesh. The extended B-splines are the extension of classical B-splines. To find the best value of the optimization technique is adopted. The extended cubic B-splines are an advantage over the classical B-splines as for some optimized value of the solution obtained by the extended B-splines is better than the solution obtained by classical B-splines. The method is shown to be first-order accurate in t and almost the second-order accurate in x. It is also shown that this method is better than some existing methods. Several test problems are encountered to validate the theoretical results. KEYWORDS delay partial differential equations, extended B-splines, free parameter, parabolic boundary layers, parameter-uniform convergence, piecewise-uniform mesh, singular perturbation, time delay Numer Methods Partial Differential Eq. 2020;36:868-886. wileyonlinelibrary.com/journal/num

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Shishkin mesh based septic Hermite interpolation algorithm for time-dependent singularly perturbed convection–diffusion models

Kumari

Kukreja

2022

J Math Chem

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Finite Element Analysis for Mass-Lumped Three-Step Taylor Galerkin Method for Time Dependent Singularly Perturbed Problems with Exponentially Fitted Splines

Cited by 7 publications

References 15 publications

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

A parameter‐uniform scheme for singularly perturbed partial differential equations with a time lag

Shishkin mesh based septic Hermite interpolation algorithm for time-dependent singularly perturbed convection–diffusion models

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