scholarly journals A uniform finite element method for a conservative singularly perturbed problem

1987 ◽  
Vol 18 (2) ◽  
pp. 163-174 ◽  
Author(s):  
Eugene O'Riordan ◽  
Martin Stynes
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Element Method

Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem

2017 ◽  
Vol 10 (1) ◽  
pp. 44-64 ◽  
Author(s):  
Yunhui Yin ◽  
Peng Zhu ◽  
Bin Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Perturbation Parameter ◽  
Diffusion Problem ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Convection Diffusion ◽  
Streamline Diffusion ◽  
Element Method

AbstractIn this paper, a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection – diffusion problem is analyzed. The method is shown to be convergent uniformly in the perturbation parameter ∈ provided only that ∈ ≤ N–1. An convergent rate in a discrete streamline-diffusion norm is established under certain regularity assumptions. Finally, through numerical experiments, we verified the theoretical results.

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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