scholarly journals On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions

1983 ◽  
Vol 40 (161) ◽  
pp. 47-47 ◽  
Author(s):  
A. H. Schatz ◽  
L. B. Wahlbin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
The Finite Element Method ◽  
Diffusion Problems ◽  
Element Method

PARAMETER-UNIFORM FINITE ELEMENT METHOD FOR TWO-PARAMETER SINGULARLY PERTURBED PARABOLIC REACTION-DIFFUSION PROBLEMS

2012 ◽  
Vol 09 (04) ◽  
pp. 1250047 ◽  
Author(s):  
M. K. KADALBAJOO ◽  
ARJUN SINGH YADAW
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Rectangular Domain ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Spatial Direction ◽  
Small Parameters ◽  
Diffusion Problems ◽  
Element Method

In this paper, parameter-uniform numerical methods for a class of singularly perturbed one-dimensional parabolic reaction-diffusion problems with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and finite element method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O(N-2( ln N)2 + Δt). Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

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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