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An iterative numerical algorithm for a strongly coupled system of singularly perturbed convection-diffusion problems. (English) Zbl 1233.65079

Margenov, Svetozar (ed.) et al., Numerical analysis and its applications. 4th international conference, NAA 2008, Lozenetz, Bulgaria, June 16–20, 2008. Revised selected papers. Berlin: Springer (ISBN 978-3-642-00463-6/pbk). Lecture Notes in Computer Science 5434, 104-115 (2009).
Summary: An iterative numerical method is constructed for a coupled system of singularly perturbed convection-diffusion-reaction two-point boundary value problems. It combines a standard finite difference operator with a piecewise-uniform Shishkin mesh, and uses a Jacobi-type iteration to compute a solution. Under certain assumptions on the coefficients in the differential equations, a bound on the maximum-norm error in the computed solution is established; this bound is independent of the values of the singular perturbation parameter. Numerical results are presented to illustrate the performance of the numerical method.
For the entire collection see [Zbl 1157.65002].

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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[1] Kellogg, R., Linß, T., Stynes, M.: A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions. Math. Comp. 77, 2085–2096 (2008) · Zbl 1198.65209
[2] Kellogg, R., Madden, N., Stynes, M.: A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions. Numer. Methods. Partial Diff. Eqns. 24, 312–334 (2008) · Zbl 1133.65088
[3] Linß, T., Madden, N.: An improved error estimate for a numerical method for a system of coupled singularly perturbed reaction–diffusion equations. Comp. Meth. Appl. Math. 3, 417–423 (2003) · Zbl 1040.65066
[4] Linß, T., Madden, N.: Accurate solution of a system of coupled singularly perturbed reaction-diffusion equations. Computing 73, 121–133 (2004) · Zbl 1057.65046
[5] Madden, N., Stynes, M.: A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems. IMA J. Numer. Anal. 23, 627–644 (2003) · Zbl 1048.65076
[6] Matthews, S., O’Riordan, E., Shishkin, G.: A numerical method for a system of singularly perturbed reaction-diffusion equations. J. Comput. Appl. Math. 145, 151–166 (2002) · Zbl 1004.65079
[7] Shishkin, G.I.: Approximation of a system of singularly perturbed elliptic reaction–diffusion equations on a rectangle. In: Farago, I., Vabishchevich, P., Vulkov, L. (eds.) Proceedings of Fourth International Conference on Finite Difference Methods: Theory and Applications, Lozenetz, Bulgaria, Rousse University, Bulgaria, August 2006, pp. 125–133 (2007)
[8] Bakhvalov, N.S.: On the optimization of methods for boundary-value problems with boundary layers. J. Numer. Meth. Math. Phys. 9(4), 841–859 (1969) (in Russian)
[9] Shishkin, G.: Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations. Comp. Maths. Math. Phy 35, 429–446 (1995) · Zbl 0852.65071
[10] Gracia, J., Lisbona, F., O’Riordan, E.: A system of singularly perturbed reaction-diffusion equations. Preprint MS-07-10, Dublin City University (2007)
[11] Linß, T., Madden, N.: Layer-adapted meshes for a system of coupled singularly perturbed reaction-diffusion problem. IMA Journal of Numerical Analysis 29, 109–125 (2009) · Zbl 1168.65046
[12] Andreev, V.: On the uniform convergence of a classical difference scheme on a nonuniform mesh for the one-dimensional singularly perturbed reaction-diffusion equation. Comp. Math. Math. Phys. 44, 449–464 (2004)
[13] Shishkina, L., Shishkin, G.: Robust numerical method for a system of singularly perturbed parabolic reaction–diffusion equations on a rectangle. Math. Model. Anal. 13, 251–261 (2008) · Zbl 1149.65074
[14] Cen, Z.: Parameter-uniform finite difference scheme for a system of coupled singularly perturbed convection-diffusion equations. J. Syst. Sci. Complex. 18, 498–510 (2005) · Zbl 1087.65072
[15] Linß, T.: Analysis of an upwind finite-difference scheme for a system of coupled singularly perturbed convection-diffusion equations. Computing 79, 23–32 (2007) · Zbl 1115.65084
[16] O’Riordan, E., Stynes, M.: Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems. Adv. Comput. Math. 30, 101–121 (2009) · Zbl 1167.65044
[17] Linß, T.: Analysis of a system of singularly perturbed convection-diffusion equations with strong coupling. Preprint MATH-NM-02-2007, Institut für Numerische Mathematik, Technische Universität Dresden (2007)
[18] O’Riordan, E., Stynes, J., Stynes, M.: A parameter-uniform finite difference method for a coupled system of convection-diffusion two-point boundary value problems. Numerical Mathematics: Theory, Methods and Applications 1, 176–197 (2008) · Zbl 1174.65441
[19] Andreev, V.: A priori estimates for solutions of singularly perturbed two-point boundary value problems. Mat. Model. 14, 5–16 (2002); Second International Conference OFEA 2001, Optimization of Finite Element Approximation, Splines and Wavelets (St. Petersburg, 2001) (Russian) · Zbl 1036.34025
[20] Ladyzhenskaya, O., Uraĺtseva, N.: Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York (1968)
[21] Farrell, P.A., Hegarty, A.F., Miller, J.J., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC, Boca Raton (2000) · Zbl 0964.65083
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