A finite element method for two–parameter singularly perturbed problems in 2D

PAMM ◽  
2006 ◽  
Vol 6 (1) ◽  
pp. 771-772 ◽  
Author(s):  
Ljiljana Teofanov ◽  
Hans-Görg Roos ◽  
Helena Zarin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
Two Parameter ◽  
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.

THE hp FINITE ELEMENT METHOD FOR SINGULARLY PERTURBED PROBLEMS IN SMOOTH DOMAINS

1998 ◽  
Vol 08 (02) ◽  
pp. 299-326 ◽  
Author(s):  
CHRISTOS A. XENOPHONTOS
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Approximation ◽  
Singularly Perturbed ◽  
Exponential Rate ◽  
Product Spaces ◽  
Singularly Perturbed Problems ◽  
Smooth Part ◽  
Element Method ◽  
Hp Finite Element

We consider the numerical approximation of singularly perturbed elliptic problems in smooth domains. The solution to such problems can be decomposed into a smooth part and a boundary layer part. We present guidelines for the effective resolution of boundary layers in the context of the hp finite element method and we construct tensor product spaces that approximate these layers uniformly at a near-exponential rate.

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