A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem

O’Riordan, Eugene; Stynes, Martin

doi:10.1090/s0025-5718-1986-0856702-7

Cited by 30 publications

(11 citation statements)

References 5 publications

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“…The proposed numerical method is accurate of order O(N −1 ln N ). Tables 6.2-6.5 and 6.7-6.8 shows, how the present method is more efficient than the methods given in [11,12,19,21]. To further corroborate the applicability of the proposed method, graphs between exact solution and approximate solutions have been plotted for the first three examples for a fixed = 10 −3 and N = 64.…”

Section: Discussionmentioning

confidence: 78%

Variable Mesh Finite Difference Method for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems

Kumar¹

2010

JCM

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A numerical method based on finite difference method with variable mesh is given for self-adjoint singularly perturbed two-point boundary value problems. To obtain parameteruniform convergence, a variable mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The uniform convergence analysis of the method is discussed. The original problem is reduced to its normal form and the reduced problem is solved by finite difference method taking variable mesh. To support the efficiency of the method, several numerical examples have been considered.Mathematics subject classification: 45Exx, 65L10, 65L12, 65L50, 65L70.

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Section: Discussionmentioning

confidence: 78%

Variable Mesh Finite Difference Method for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems

Kumar¹

2010

JCM

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“…In another great study [2], Parul gave some examples to these kinds of problems occuring in almost all science branches and briefly examined the conventional methods. Various numerical approaches also were applied to approximate to the solutions of turning point problems such as finite difference methods [5], finite element method [6,7], numerical integration method [11], initial value techniques [12] and Reproducing Kernel Method [13,14]. In the detailed work [15], Sharma et al review the theory and methods, and study the development covering the years 1970-2011.…”

Section: Introductionmentioning

confidence: 99%

“…Some of them can be given as : finite difference methods [5], finite element methods [6,7], the Method of c ⃝ 2016 BISKA Bilisim Technology (WKB) Approximation [8,9], the Method of Matched Asymptotic Expansions (MMAE) [10], numerical integration methods [11], initial value techniques [12] and reproducing kernel methods (RKM) [13,14]. In the detailed work [15], Sharma et al review the theory and methods, and study the development covering the years 1970-2011.…”

Section: Introductionmentioning

confidence: 99%

SCEM approach for singularly perturbed linear turning mid-point problems with an interior layer

Atay¹,

Cengizci²,

Eryılmaz³

2016

ntmsci

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Abstract:In this work, we study an important class of singular perturbation problems: singularly perturbed two-point turning point problems (TPP's) in ordinary differential equations (ODE's) exhibiting interior layer at the mid-point of their intervals. We consider this type of problems as two different singular perturbation problems which have a boundary layer at an end point of their intervals. This study is devoted to the Successive complementary expansion method (SCEM) to solve these two sub-problems. Two test problems are considered to check the efficiency and accuracy of the proposed method. Our numerical experiments show that SCEM approximations are in good agreement with exact and previously obtained solutions.

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“…There is an extensive literature on numerical methods for singularly perturbed reaction diffusion problems (see, for example, [1,10,12,17] and the references therein). Our interest lies in examining parameter-uniform numerical methods [3,8] for singularly perturbed problems.…”

Section: Introductionmentioning

confidence: 99%

A parameter robust numerical method for a two dimensional reaction-diffusion problem

2005

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Abstract. In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.

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A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem

Cited by 30 publications

References 5 publications

Variable Mesh Finite Difference Method for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems

Variable Mesh Finite Difference Method for Self-Adjoint Singularly Perturbed Two-Point Boundary Value Problems

SCEM approach for singularly perturbed linear turning mid-point problems with an interior layer

A parameter robust numerical method for a two dimensional reaction-diffusion problem

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