scholarly journals Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem

2008 ◽  
Vol 77 (261) ◽  
pp. 21-39 ◽  
Author(s):  
Dmitriy Leykekhman
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Uniform Error ◽  
Uniform Error Estimates ◽  
Element Method

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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