A singularly perturbed problem with two parameters in two dimensions on graded meshes

Brdar, Mirjana; Zarin, Helena; Teofanov, Ljiljana

doi:10.1016/j.camwa.2016.09.021

Cited by 14 publications

(9 citation statements)

References 14 publications

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“…However, the accuracy of the modified graded mesh prevails in the case p = 1 in comparison with Bakhvalov mesh. Overall, the results obtained over modified graded mesh are better than those presented in Brdar et al 22…”

Section: Numerical Results and Discussioncontrasting

confidence: 63%

“…32 Example 6.1. Consider the following two-parameter singularly perturbed boundary value problem 31,32 : Table 3 present a comparison of energy norm error for the Galerkin finite element method on Bakhvalov mesh 22 and modified graded mesh for a predefined range of parameters. This table exhibits the results obtained by using linear and quadratic polynomials in the finite element basis.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…In Brdar et al, 22 a two-parameter problem is studied over a graded mesh. These meshes are an effective alternative to Shishkin class of meshes.…”

Section: Introductionmentioning

confidence: 99%

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A higher order finite element method with modified graded mesh for singularly perturbed two‐parameter problems

Kaushik

Kumar

Sharma³

et al. 2020

Math Methods in App Sciences

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This paper presents a modified graded mesh for singularly perturbed two‐parameter problems. The mesh is generated recursively using Newton's algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on higher order polynomials of degree p≥1. We prove parameter uniform convergence of optimal order in ε‐weighted energy norm. A test example is taken to compare the proposed graded mesh with others found in the literature.

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Section: Numerical Results and Discussioncontrasting

confidence: 63%

Section: Numerical Results and Discussionmentioning

confidence: 99%

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A higher order finite element method with modified graded mesh for singularly perturbed two‐parameter problems

Kaushik

Kumar

Sharma³

et al. 2020

Math Methods in App Sciences

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show abstract

“…The usage of graded mesh is mainly important in finite element (FEM) [14][15][16][17][18], finite-difference (FDM) [19], exponential B-spline [14], and Newton methods [18]. In particular, the mesh is highly useful in the numerical experiment of reaction-diffusion problems [14,16,17], singularly perturbed problems with two parameters [15], sub-diffusion problems with nonlocal diffusion term [18], and evolution problems with a weakly singular kernel [19].…”

Section: Mesh Challengesmentioning

confidence: 99%

Effect of Graded Mesh Number on the Solution of Convection-Diffusion Flow Problem with Quadratic Source

Abdullah

2023

ARFMTS

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Convection-diffusion phenomena are fundamentally modelled in a variety of engineering fields and physical sciences. A suitable meshing strategy is needed in computational fluid dynamics to solve model problems numerically. Unintentional implementation of the approach could lead to subpar solutions such as erroneous oscillations, over- or under-predictions, and excessive computing time. This paper highlights the significance of the influence of mesh structure on the solution of convection-diffusion flow problem with quadratic source for flow parameters of interest. Particularly, it presents the accuracy of the solution of the flow problem at low Peclet number with respect to graded mesh number, where the expansion factor used is based on an established linear logarithmic model involving Peclet number derived in previous works. The problem is solved by assigning several mesh intervals to graded mesh against each Peclet number of interest. Based on the values of the interval and Peclet number, 16 test cases are considered. Quantitative results lead to orders of accuracy of the solution of the flow problem. The effect of graded mesh number on the accuracy thus serves as a reference for a more structured decision-making and improves the heuristic process in choosing the computational domain mesh with expected order of accuracy for the numerical solution of the problem particularly in the calculation of scalar concentration. The orders of accuracy confirm the profiles of concentration.

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“…Here we only discuss the case of 0 < ε 1 , ε 2 ≪ 1 (see [7]), and if you are interested in the case of ε 2 = 0 and ε 2 = 1, you can refer to [12] and its references. For the sake of analysis, we use b(x) and c(x) in our problem instead of b(x, y) and c(x, y), because that doesn't affect the properties of the true solution(see [10]), and this is also the case in the equations of [11], [12], and [4].…”

Section: Introductionmentioning

confidence: 99%

Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D

Zhang¹,

Lv²

2020

Preprint

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For a singularly perturbed elliptic model problem with two small parameters, we analyze finite element methods of any order on a Bakhvalov-type mesh.For convergence analysis, we construct a new interpolation by using the characteristics of layers. Besides, a more subtle analysis of the mesh scale near the exponential layer is carried out. Based on the interpolation and new analysis of the mesh scale, we prove the optimal convergence order.

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A singularly perturbed problem with two parameters in two dimensions on graded meshes

Cited by 14 publications

References 14 publications

A higher order finite element method with modified graded mesh for singularly perturbed two‐parameter problems

A higher order finite element method with modified graded mesh for singularly perturbed two‐parameter problems

Effect of Graded Mesh Number on the Solution of Convection-Diffusion Flow Problem with Quadratic Source

Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D

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