A posteriori adaptive mesh technique with a priori error estimates for singularly perturbed semilinear parabolic convection-diffusion equations

2007 ◽  
Vol 1 (2/3/4) ◽  
pp. 374
Author(s):  
G.I. Shishkin
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
Adaptive Mesh ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
A Posteriori ◽  
Convection Diffusion ◽  
Priori Error Estimates ◽  
Semilinear Parabolic ◽  
Mesh Technique

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’.

scholarly journals A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yang Liu ◽  
Hong Li ◽  
Wei Gao ◽  
Siriguleng He ◽  
Zhichao Fang
Keyword(s):  
Error Estimates ◽  
Mixed Method ◽  
Mixed Finite Element ◽  
A Priori ◽  
A Priori Error Estimates ◽  
The Novel ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Priori Error Estimates ◽  
Diffusion Problems

A novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term∇·(a(x,t)∇u)is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classicalH(div;Ω)space and the hyperbolic partd(x)(∂u/∂t)+c(x,t)·∇uis handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates inL2- andH1-norms for the scalar unknownuand a priori error estimates in(L2)2-norm for its gradientλand its fluxσ(the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.

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