×

Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities. (English) Zbl 1244.65173

The authors consider a 2D diffusion-convection problem with a small \(\epsilon\) multiplying the Laplace operator, and with smooth, positive \(p\), \(q\) multiplying \(u_x\) and \(u\). Along with this, Dirichlet conditions are given on the boundary of the unit square. For the numerical solution, they investigate the Galerkin, resp. streamline diffusion finite element method on a Shishkin grid and taking bilinear elements. The main body of the paper is devoted to accuracy estimates, uniformly in \(\epsilon\), which are obtained by assuming that the same decomposition of the exact solution holds as for constants \(p\), \(q\) (containing exponential and parabolic boundary layers, and corner singularities). A careful analysis shows an estimate \(N^{-1}\cdot\ln N\) - which is confirmed to hold by numerical experiments.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Thomas Apel, Anisotropic finite elements: local estimates and applications, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1999. · Zbl 0917.65090
[2] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Robust computational techniques for boundary layers, Applied Mathematics (Boca Raton), vol. 16, Chapman & Hall/CRC, Boca Raton, FL, 2000. · Zbl 0964.65083
[3] Sebastian Franz and Torsten Linß, Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers, Numer. Methods Partial Differential Equations 24 (2008), no. 1, 144 – 164. · Zbl 1133.65090 · doi:10.1002/num.20245
[4] S. Franz, T. Linß, and H.-G. Roos, Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers, Appl. Numer. Math. 58 (2008), no. 12, 1818 – 1829. · Zbl 1221.65292 · doi:10.1016/j.apnum.2007.11.005
[5] R. Bruce Kellogg and Martin Stynes, Corner singularities and boundary layers in a simple convection-diffusion problem, J. Differential Equations 213 (2005), no. 1, 81 – 120. · Zbl 1159.35309 · doi:10.1016/j.jde.2005.02.011
[6] R. Bruce Kellogg and Martin Stynes, Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem, Appl. Math. Lett. 20 (2007), no. 5, 539 – 544. · Zbl 1201.35094 · doi:10.1016/j.aml.2006.08.001
[7] Q. Lin, A rectangle test for finite element analysis, Proc. Syst. Sci. Eng., Great Wall (H.K.) Culture Publish Co., 1991, pp. 213-216.
[8] Torsten Linß, Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes, Numer. Methods Partial Differential Equations 16 (2000), no. 5, 426 – 440. , https://doi.org/10.1002/1098-2426(200009)16:53.3.CO;2-I · Zbl 0958.65110
[9] T. Linßand M. Stynes, Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 190 (2001), 3527-3542. · Zbl 0988.76062
[10] Hans-Görg Roos, Martin Stynes, and Lutz Tobiska, Robust numerical methods for singularly perturbed differential equations, 2nd ed., Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 2008. Convection-diffusion-reaction and flow problems. · Zbl 1155.65087
[11] Grigory I. Shishkin and Lidia P. Shishkina, Difference methods for singular perturbation problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 140, CRC Press, Boca Raton, FL, 2009. · Zbl 1163.65062
[12] Martin Stynes and Lutz Tobiska, The SDFEM for a convection-diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy, SIAM J. Numer. Anal. 41 (2003), no. 5, 1620 – 1642. · Zbl 1055.65121 · doi:10.1137/S0036142902404728
[13] Zhimin Zhang, Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems, Math. Comp. 72 (2003), no. 243, 1147 – 1177. · Zbl 1019.65091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.