An adapted Petrov–Galerkin multi-scale finite element method for singularly perturbed reaction–diffusion problems

2015 ◽  
Vol 93 (7) ◽  
pp. 1200-1211 ◽  
Author(s):  
Shan Jiang ◽  
Michael Presho ◽  
Yunqing Huang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Multi Scale ◽  
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.

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