Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with characteristic layers
type: Item
instance of: scholarly article
title: Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with characteristic layers
zbMATH Open document ID: 1397.65280
author: Xiaowei Liu
author: Jin Zhang
published in: Numerical Algorithms
publication date: 2018-06-25T00:00:00Z
review text: A Shishkin mesh is a piecewise uniform mesh (or a tensor-product version in more than one dimension). What distinguishes a Shishkin mesh from any other piecewise uniform mesh is the choice of the so-called transition parameter(s), which are the point(s) at wich the mesh size changes abruptly. A different approach is to use layer-adapted meshes. Among these, we cite Shishkin meshes (see [\textit{T. J. R. Hughes} and \textit{A. Brooks}, in: Finite element methods for convection dominated flows, Winter ann. Meet. ASME, New York 1979, AMD Vol. 34, 19--35 (1979; Zbl 0423.76067)], [\textit{G. I. Shishkin}, Grid approximation of singularly perturbed elliptic and parabolic equations. Ekaterinburg: Russian Academy of Sciences, Ural Branch; Moscow: Keldysh Institute of Applied Mathematics, RAS (Habilitation 1991). 232~p. (1992; Zbl 1397.65005)]. On the basis of the following problem: \[ -\varepsilon\Delta u+bu_x+cu=f\text{ in }\Omega=(0,1)^2, \] \(u=0\) on \(\partial\Omega\), where \(b,c>0\) are constants, \(b\geq\beta\) on \(\Omega\), \(\beta\) a positive constant, \(\varepsilon\) a small positive parameter with \(\varepsilon\ll b\) and f is sufficiently smooth. The solution of the Problem (1) has an exponential layer of width \(O(\varepsilon\ln(1/\varepsilon))\) near the outflow boundary at \(x=1\) and two parabolic layers of width \(O(\sqrt \varepsilon\ln((1/\varepsilon)))\) near the characteristic boundaries at \(y=0\) and \(y=1\). The authors develop their method using discrete Green's functions for different kinds of FEs. For SDFEMs on quasiuniform meshes ([\textit{A. H. Schatz} and \textit{L. B. Wahlbin}, Math. Comput. 33, 465--492 (1979; Zbl 0417.65053); ibid. 32, 73--109 (1978; Zbl 0382.65058)], [\textit{A. H. Schatz}, Math. Comput. 67, No. 223, 877--899 (1998; Zbl 0905.65105)], [\textit{K. Niijima}, Numer. Math. 56, No. 7, 707--719 (1990; Zbl 0691.65077)], [\textit{G. Zhou}, Math. Comput. 66, No. 217, 31--44 (1997; Zbl 0854.65094)]) and on Shishkin meshes ([\textit{…
reviewed by: Noureddine Daili
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65N50
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65M25
zbMATH DE Number: 6894096
zbMATH Keywords: convection-diffusion
zbMATH Keywords: characteristic layers
zbMATH Keywords: Shishkin triangular mesh
zbMATH Keywords: SDFEM
zbMATH Keywords: pointwise error
MaRDI profile type: Publication
full work available at URL: s11075-017-0384-z
OpenAlex ID: W2740484553
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DOI: 10.1007/S11075-017-0384-Z
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description: scientific article; zbMATH DE number 6894096