An analysis of a conforming exponentially fitted finite element method for a convection-diffusion problem
type: Item
instance of: scholarly article
title: An analysis of a conforming exponentially fitted finite element method for a convection-diffusion problem
zbMATH Open document ID: 1008.65081
DOI: 10.1016/S0377-0427(01)00530-1
published in: Journal of Computational and Applied Mathematics
publication date: 2002-08-22T00:00:00Z
review text: This very interesting and important paper is devoted to theory of projective-grid methods for the stationary linear convection-diffusion equation in a polygonal region \(\Omega\) under the standard Dirichlet conditions. The equation is written as \(-\nabla\cdot [\varepsilon\nabla u-u\vec au]+gu=f\) and it is assumed that: \(\vec a\in [C^1(\overline\Omega)]^2\), \(\nabla\vec a +2g\geq 0 \), \(g\in L^{\infty}(\Omega)\), \(f\in L_2(\Omega)\), and that the solution \(u\in H^2(\Omega)\), Then \(\|u\|_1=O(\varepsilon^{-1/2})\) and \(\|u\|_2=O(\varepsilon^{-3/2})\). To obtain effective approximations (in the presence of the boundary layer) special projective-grid methods (the Galerkin finite element methods) are used on the base of triangulations \(T_h(\overline \Omega)\). The approximating subspaces instead of the standard piecewise linear splines make use of special more involved functions on the triangular cells via exponential fitting in dependence of the given vector \(\vec a\). In the case of regular triangulations \(T_h(\overline \Omega)\) estimates of errors in the energy norm are given of the form \(\|u-u_h\|_E=O(h\varepsilon^{-1/2})\). In case of the unit square and \(\vec a\equiv[a_1,a_2]\) with \(a_i\asymp 1\) and a layer-fitted grids with refinement, the authors obtained remarkable estimates of the type \(\|u-u_h\|_E=O(h^{1/2}[1+h^{1/2}|\ln \varepsilon])\) almost independent of the singular parameter. The number of the triangles is \(O(h^{-2})\).
reviewed by: Evgeniĭ Dyakonov
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N50
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1787642
zbMATH Keywords: convection-diffusion problems
zbMATH Keywords: projective-grid methods
zbMATH Keywords: Galerkin finite element methods
zbMATH Keywords: exponential fitting
zbMATH Keywords: boundary layer
author: Song Wang
author: Zi-Cai Li
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787642