Finite element methods for a singularly perturbed transmission problem

2009 ◽  
Vol 17 (4) ◽  
Author(s):  
S. Nicaise ◽  
C. Xenophontos
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Transmission Problem ◽  
Singularly Perturbed

Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems

2019 ◽  
Vol 27 (1) ◽  
pp. 37-55 ◽  
Author(s):  
Stephen Russell ◽  
Martin Stynes
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Two Dimensions ◽  
Sparse Grid ◽  
Singularly Perturbed ◽  
Balanced Norm ◽  
Norm Error ◽  
Diffusion Problems

Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.

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