scholarly journals The Midpoint Upwind Scheme on the Bakhvalov-Shishkin Mesh for Parabolic Singularly Perturbed Problems

2021 ◽  
Vol 2012 (1) ◽  
pp. 012047
Author(s):  
Quan Zheng ◽  
Yi-Yang Wang
Keyword(s):  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Upwind Scheme ◽  
Singularly Perturbed Problems

ε-Uniform Convergence of the Hybrid Difference Scheme on the Shishkin Mesh for Singularly Perturbed Problems

2013 ◽  
Vol 392 ◽  
pp. 214-217 ◽  
Author(s):  
Quan Zheng ◽  
Xiao Li Feng ◽  
Xue Zheng Li
Keyword(s):  
Theoretical Result ◽  
Difference Scheme ◽  
Uniform Convergence ◽  
Comparison Principle ◽  
Pointwise Convergence ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
Barrier Functions

In this paper, we discuss the hybrid difference scheme on the Shishkin mesh for the singularly perturbed problems in 2-D. The ε-uniform pointwise convergence is proved by the comparison principle and barrier functions. The experiments support the theoretical result.

Regularization of Integral Operators in Singularly Perturbed Problems Using Normal Forms

Vestnik MEI ◽  
2019 ◽  
Vol 6 ◽  
pp. 131-137
Author(s):  
Abdukhafiz A. Bobodzhanova ◽  
◽  
Valeriy F. Safonov ◽  
Keyword(s):  
Normal Forms ◽  
Integral Operators ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems

An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data

2021 ◽  
pp. 2150056
Author(s):  
Sheetal Chawla ◽  
Jagbir Singh ◽  
Urmil
Keyword(s):  
Uniform Convergence ◽  
Order Parameter ◽  
Source Term ◽  
Reaction Diffusion ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Diffusion Type ◽  
Bakhvalov Mesh ◽  
Second Order Parameter

In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

A Parameter Robust Finite Element Method for Fourth Order Singularly Perturbed Problems

2017 ◽  
Vol 17 (2) ◽  
pp. 337-349 ◽  
Author(s):  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Hermite Polynomials ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Singularly Perturbed Problems ◽  
Exponentially Graded ◽  
Element Method

AbstractWe consider fourth order singularly perturbed problems in one-dimension and the approximation of their solution by the h version of the finite element method. In particular, we use piecewise Hermite polynomials of degree ${p\geq 3}$ defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error is measured in both the energy norm and a stronger, ‘balanced’ norm. Finally, we illustrate our theoretical findings through numerical computations, including a comparison with another scheme from the literature.

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