Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with characteristic layers
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{T. Linß} and \textit{M. Stynes}, Comput. Methods Appl. Mech. Eng. 190, No. 28, 3527--3542 (2001; Zbl 0988.76062); Numer. Math. 87, No. 3, 457--484 (2001; Zbl 0969.65106)], [\textit{J. Zhang} et al., Appl. Numer. Math. 64, 19--34 (2013; Zbl 1255.65214)]). To illustrate the theory given in the present study and examine the performance of the proposed numerical scheme numerical experiments are carried out. Theoretical analysis is carried out in order to obtain the stability and error estimate. It is proved that uniformly pointwise error bounds away from the layers are of order almost 7/4 using a variant of artificial crosswind diffusion. In some cases, the convergence order is almost 15/8. Unfortunately, only one numerical example is not enough. One needs a broad class of examples and comparison to give conclusion on the effectiveness of the results.
scientific article; zbMATH DE number 6894096
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