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A second-order overlapping Schwarz method for a 2D singularly perturbed semilinear reaction-diffusion problem

Authors: Natalia Kopteva and Maria Pickett
Journal: Math. Comp. 81 (2012), 81-105
MSC (2010): Primary 65N06, 65N15, 65N30, 65N50, 65N55
Published electronically: July 18, 2011
MathSciNet review: 2833488
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Abstract: An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion equation posed in a smooth two-dimensional domain. The problem may exhibit multiple solutions; its diffusion parameter $ \varepsilon^2$ is arbitrarily small, which induces boundary layers. The Schwarz method invokes a boundary-layer subdomain and an interior subdomain, the narrow subdomain overlap being of width $ O(\varepsilon\vert\ln h\vert)$, where $ h$ is the maximum side length of mesh elements, and the global number of mesh nodes does not exceed $ O(h^{-2})$. We employ finite differences on layer-adapted meshes of Bakhvalov and Shishkin types in the boundary-layer subdomain, and lumped-mass linear finite elements on a quasiuniform Delaunay triangulation in the interior subdomain. For this iterative method, we present maximum norm error estimates for $ \varepsilon\in(0,1]$. It is shown, in particular, that when $ \varepsilon\le C\vert\ln h\vert^{-1}$, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in $ \varepsilon$. Numerical results are presented to support our theoretical conclusions.

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  • 1. V. B. Andreev and N. Kopteva, Pointwise approximation of corner singularities for a singularly perturbed reaction-diffusion equation in an $ L$-shaped domain, Math. Comp., 77 (2008), 2125-2139. MR 2429877 (2009d:65143)
  • 2. N. S. Bakhvalov, On the optimization of methods for solving boundary value problems with boundary layers, Zh. Vychisl. Mat. Mat. Fis., 9 (1969), 841-859 (in Russian). MR 0255066 (40:8273)
  • 3. I. A. Blatov, Galerkin finite element method for elliptic quasilinear singularly perturbed boundary problems. I, (Russian) Differ. Uravn., 28 (1992), 1168-1177; translation in Differ. Equ., 28 (1992), 931-940. MR 1201213 (94a:65056)
  • 4. C. Clavero, J. L. Gracia and E. O'Riordan, A parameter robust numerical method for a two dimensional reaction-diffusion problem, Math. Comp., 74 (2005), 1743-1758. MR 2164094 (2006e:65192)
  • 5. P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232. MR 0374665 (51:10863)
  • 6. B. Heinrich and K. Pönitz, Nitsche type mortaring for singularly perturbed reaction-diffusion problems, Computing, 75 (2005), 257-279. MR 2173513 (2006g:65185)
  • 7. R. B. Kellogg and N. Kopteva, A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain, J. Differential Equations, 248 (2010), 184-208. MR 2557900 (2011a:35178)
  • 8. N. Kopteva, Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem, Math. Comp., 76 (2007), 631-646. MR 2291831 (2008e:65327)
  • 9. N. Kopteva, N. Madden and M. Stynes, Grid equidistribution for reaction-diffusion problems in one dimension, Numer. Algorithms, 40 (2005), 305-322. MR 2189409 (2006h:65099)
  • 10. N. Kopteva, M. Pickett and H. Purtill, A robust overlapping Schwarz method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Int. J. Numer. Anal. Model., 6 (2009), 680-695. MR 2574759 (2010m:65155)
  • 11. N. Kopteva and M. Stynes, Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions, Appl. Numer. Math., 51 (2004), 273-288. MR 2091404 (2005e:65097)
  • 12. H.-O. Kreiss, T. A. Manteuffel, B. Swartz, B. Wendroff and A. B. White, Jr., Supra-convergent schemes on irregular grids, Math. Comp., 47 (1986), 537-554. MR 856701 (88b:65082)
  • 13. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. MR 0244627 (39:5941)
  • 14. D. Leykekhman, Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem, Math. Comp., 77 (2008), 21-39. MR 2353942 (2009b:65316)
  • 15. J. Lorenz, Nonlinear singular perturbation problems and the Enquist-Osher scheme, Report 8115, Mathematical Institute, Catholic Univerity of Nijmegen, 1981 (unpublished).
  • 16. H. MacMullen, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffustion problems with boundary layers, J. Comput. Appl. Math., 130 (2001), 231-244. MR 1827983 (2002c:65113)
  • 17. J. M. Melenk, $ hp$-finite element methods for singular perturbations, Springer, 2002. MR 1939620 (2003i:65108)
  • 18. N. N. Nefedov, The method of differential inequalities for some classes of nonlinear singularly perturbed problems with internal layers, (Russian) Differ. Uravn., 31 (1995), 1142-1149; translation in Differ. Equ., 31 (1995), 1077-1085. MR 1429769 (97m:35018)
  • 19. C. V. Pao, Monotone iterative methods for finite difference system of reaction-diffusion equations, Numer. Math., 46 (1985), 571-586. MR 796645 (86h:65156)
  • 20. A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, Clarendon Press, Oxford, 1999. MR 1857663 (2002i:65002)
  • 21. H.-G. Roos, A note on the conditioning of upwind schemes on Shishkin meshes, IMA J. Numer. Anal., 16 (1996), 529-538. MR 1414845 (97g:65204)
  • 22. A. A. Samarskii, Theory of Difference Schemes, Nauka, Moscow, 1989 (in Russian). MR 1196231 (93g:65004)
  • 23. A. H. Schatz and L. B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp., 40 (1983), 47-89. MR 679434 (84c:65137)
  • 24. G. I. Shishkin, Grid approximation of singularly perturbed elliptic and parabolic equations, Ur. O. Ran, Ekaterinburg, 1992 (in Russian).
  • 25. M. Stephens and N. Madden, A parameter-uniform Schwarz method for a coupled system of reaction-diffusion equations, J. Comput. Appl. Math., 230 (2009), 360-370. MR 2532330 (2010d:65167)
  • 26. G. Strang and G. J. Fix, An analysis of the finite element method, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR 0443377 (56:1747)
  • 27. G. Sun and M. Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions, Math. Comp., 65 (1996), 1085-1109. MR 1351205 (96j:65067)
  • 28. E. Süli, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes, SIAM J. Numer. Anal., 28 (1991), 1419-1430. MR 1119276 (92h:65159)
  • 29. V. Thomée, J.-C. Xu and N.-Y. Zhang, Superconvergence of the gradient in piecewise linear finite-element approximation to a parabolic problem, SIAM J. Numer. Anal., 26 (1989), 553-573. MR 997656 (90e:65165)
  • 30. J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp., 68 (1999), 1429-1446. MR 1654022 (99m:65225)

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Additional Information

Natalia Kopteva
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland

Maria Pickett
Affiliation: Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
Address at time of publication: Department of Mathematics, Lion Gate Building, Lion Terrace, Portsmouth, Hampshire PO1 3HF, United Kingdom

Keywords: Semilinear reaction-diffusion, singular perturbation, domain decomposition, overlapping Schwarz, Bakhvalov mesh, Shishkin mesh, supra-convergence, lumped-mass finite elements
Received by editor(s): December 11, 2009
Received by editor(s) in revised form: November 4, 2010
Published electronically: July 18, 2011
Additional Notes: This research was supported by an Irish Research Council for Science and Technology (IRCSET) postdoctoral fellowship and a Science Foundation Ireland grant under the Research Frontiers Programme 2008.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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