scholarly journals A linearised singularly perturbed convection–diffusion problem with an interior layer

2015 ◽  
Vol 98 ◽  
pp. 1-17 ◽  
Author(s):  
E. O'Riordan ◽  
J. Quinn
Keyword(s):  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Interior Layer ◽  
Convection Diffusion

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

2004 ◽  
Vol 4 (1) ◽  
pp. 105-127 ◽  
Author(s):  
Grigory I. Shishkin ◽  
Lidia P. Shishkina ◽  
Pieter W. Hemker
Keyword(s):  
Coordinate System ◽  
Transition Layer ◽  
A Priori ◽  
Adaptive Mesh ◽  
Singularly Perturbed ◽  
Finite Difference Schemes ◽  
A Posteriori ◽  
Interior Layer ◽  
Convection Diffusion ◽  
Mesh Technique

AbstractWe study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’.

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