Superconvergence of conforming finite element for fourth-order singularly perturbed problems of reaction diffusion type in 1D

2013 ◽  
Vol 30 (2) ◽  
pp. 550-566 ◽  
Author(s):  
Hailong Guo ◽  
Can Huang ◽  
Zhimin Zhang
Keyword(s):  
Finite Element ◽  
Fourth Order ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
Diffusion Type ◽  
Conforming Finite Element

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.

Finite Element Analysis of an Exponentially Graded Mesh for Singularly Perturbed Problems

2015 ◽  
Vol 15 (2) ◽  
pp. 135-143 ◽  
Author(s):  
Philippos Constantinou ◽  
Christos Xenophontos
Keyword(s):  
Finite Element ◽  
Reaction Diffusion ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Optimal Order ◽  
Singularly Perturbed Problems ◽  
Element Analysis ◽  
Convection Diffusion Equation ◽  
Natural Energy ◽  
Exponentially Graded

AbstractWe present the mathematical analysis for the convergence of an h version Finite Element Method (FEM) with piecewise polynomials of degree p ≥ 1, defined on an exponentially graded mesh. The analysis is presented for a singularly perturbed reaction-diffusion and a convection-diffusion equation in one dimension. We prove convergence of optimal order and independent of the singular perturbation parameter, when the error is measured in the natural energy norm associated with each problem. Numerical results comparing the exponential mesh with the Bakhvalov–Shishkin mesh from the literature are also presented.

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