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Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems

Stephen Russell and Martin Stynes

Abstract

We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.

JEL Classification: 65N12; 65N15; 65N30

  1. Funding: The research of the second author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401.

References

[1] V. B. Andreev, On the accuracy of grid approximations of nonsmooth solutions of a singularly perturbed reaction- diffusion equation in the square, Differ. Equ. 42 (2006), No. 7, 954–966.10.1134/S0012266106070044 Search in Google Scholar

[2] T. Apel, Anisotropic Finite Elements: Local Estimates and Applications, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1999. Search in Google Scholar

[3] N. Chegini and R. Stevenson, The adaptive tensor product wavelet scheme: sparse matrices and the application to singularly perturbed problems, IMA J. Numer. Anal. 32 (2012), No.1, 75–104.10.1093/imanum/drr013 Search in Google Scholar

[4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North- Holland Publishing Co., Amsterdam, 1978. Search in Google Scholar

[5] 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), No. 252, 1743–1758.10.1090/S0025-5718-05-01762-X Search in Google Scholar

[6] S. Franz, F. Liu, H.-G. Roos, M. Stynes, and A. Zhou, The combination technique for a two-dimensional convection–diffusion problem with exponential layers, Appl. Math. 54 (2009), No. 3, 203–223.10.1007/s10492-009-0013-9 Search in Google Scholar

[7] R. L.Graham, D. E.Knuth, and O.Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed, Addison-Wesley Publishing Company, Reading, MA, 1994. Search in Google Scholar

[8] H. Han and R. B. Kellogg, Differentiability properties of solutions of the equation −ε2Δu + ru = f(x, y) in a square, SIAM J. Math. Anal. 21 (1990), No. 2, 394–408.10.1137/0521022 Search in Google Scholar

[9] R. B. Kellogg, N. Madden, and M. Stynes, A parameter-robust numerical method for a system of reaction–diffusion equations in two dimensions, Numer. Methods Partial Differential Equations24 (2008), No. 1, 312–334.10.1002/num.20265 Search in Google Scholar

[10] J. Li and I. M. Navon, Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems, I. Reaction–diffusion type, Comput. Math. Appl. 35 (1998), No. 3, 57–70.10.1016/S0898-1221(97)00279-4 Search in Google Scholar

[11] R. Lin and M. Stynes, A balanced finite element method for singularly perturbed reaction–diffusion problems, SIAM J. Numer. Anal. 50 (2012), No. 5, 2729–2743.10.1137/110837784 Search in Google Scholar

[12] T. Linß, Layer-Adapted Meshes for ReactionConvectionDiffusion Problems, Lecture Notes in Mathematics, Vol. 1985, Springer-Verlag, Berlin, 2010. Search in Google Scholar

[13] F. Liu, N. Madden, M. Stynes, and A. Zhou, A two-scale sparse grid method for a singularly perturbed reaction–diffusion problem in two dimensions, IMA J. Numer. Anal. 29 (2009), No. 4, 986–1007.10.1093/imanum/drn048 Search in Google Scholar

[14] N. Madden and S. Russell, A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction–diffusion problem, Adv. Comput. Math. 41 (2015), No. 6, 987–1014.10.1007/s10444-014-9395-7 Search in Google Scholar

[15] J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. Search in Google Scholar

[16] J. Noordmans and P. W. Hemker, Application of an adaptive sparse-grid technique to a model singular perturbation problem, Computing65 (2000), No. 4, 357–378.10.1007/s006070070005 Search in Google Scholar

[17] P. Oswald, L-Bounds for the L2-Projection Onto Linear Spline Spaces, Recent Advances in Harmonic Analysis and Applications, Springer Proc. Math. Stat., Vol. 25, Springer, New York, 2013, pp. 303–316. Search in Google Scholar

[18] H.-G. Roos, Robust numerical methods for singularly perturbed differential equations: a survey covering 2008–2012, ISRN Appl. Math. (2012), Art. ID 379547, 30 p. Search in Google Scholar

[19] H.-G. Roos and M. Schopf, Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction–diffusion problems, ZAMM Z. Angew. Math. Mech. 95 (2015), No. 6, 551–565.10.1002/zamm.201300226 Search in Google Scholar

[20] H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd ed, Springer Series in Computational Mathematics, Vol. 24, Springer-Verlag, Berlin, 2008. Search in Google Scholar

[21] S. Russell and N. Madden, An introduction to the analysis and implementation of sparse grid finite element methods, Comput. Methods Appl. Math. 17 (2017), No. 2, 299–322. Search in Google Scholar

[22] G. I.Shishkin, Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural Division, Ekaterinburg, 1992 (in Russian). Search in Google Scholar

Received: 2017-06-21
Revised: 2017-10-28
Accepted: 2017-10-30
Published Online: 2018-02-01
Published in Print: 2019-03-26

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