Abstract
Collocation with quadratic C 1-splines for a singularly perturbed reaction-diffusion problem in one dimension is studied. A modified Shishkin mesh is used to resolve the layers. The resulting method is shown to be almost second order accurate in the maximum norm, uniformly in the perturbation parameter. Furthermore, a posteriori error bounds are derived for the collocation method on arbitrary meshes. These bounds are used to drive an adaptive mesh moving algorithm. Numerical results are presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of boundary layers. Z. Vycisl. Mat. Mat. Fiz. 9, 841–859 (1969)
Chadha, N.M., Kopteva, N.V.: A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem. IMA J. Numer. Anal. 31(3), 188–211 (2011)
Christara, C.C.: Quadratic spline collocation methods for elliptic partial differential equations. BIT 34(1), 33–61 (1994)
de Boor, C.: Good approximation by splines with variable knots. In: Meir, A., Sharma, A. (eds.) Spline Functions Approx. Theory, Proc. Sympos. Univ. Alberta, Edmonton 1972, pp. 57–72. Birkhäuser, Basel and Stuttgart (1973)
de Boor, C., Swartz, B.: Collocation at gaussian points. SIAM J. Numer. Anal. 10, 582–606 (1973)
Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust computational techniques for boundary layers. In: Applied Mathematics and Mathematical Computation, vol. 16. Chapman & Hall/CRC Press, Boca Raton, FL (2000)
Funaro, D.: Polynomial approximation of differential equations. In: Lecture Notes in Physics, vol. m8. Springer, Berlin (1992)
Kadalbajoo, M.K., Gupta, V., Awasthi, A.: A uniformly convergent B-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convection-diffusion problem. J. Comput. Appl. Math. 220, 271–289 (2008)
Kammerer, W.J., Reddien, G.W., Varga, R.S.: Quadratic interpolatory splines. Numer. Math. 22, 241–259 (1974)
Kopteva, N.: Maximum norm a posteriori error estimates for a 1D singularly perturbed semilinear reaction-diffusion problem. IMA J. Numer. Anal. 27(3), 576–592 (2007)
Kopteva, N.: Maximum norm a posteriori error estimate for a 2D singularly perturbed semilinear reaction-diffusion problem. SIAM J. Numer. Anal. 46(3), 1602–1618 (2008)
Kopteva, N., Stynes, M.: A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem. SIAM J. Numer. Anal. 39(4), 1446–1467 (2001)
Linß, T.: Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem. BIT Numer. Math. 47(2), 379–391 (2007)
Linß, T.: Layer-adapted meshes for reaction-convection-diffusion problems. In: Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010)
Marsden, M.J.: Quadratic spline interpolation. Bull. Am. Math. Soc. 80, 903–906 (1974)
Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted Numerical Methods for Singular Perturbation Problems. Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996)
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. In: Classics in Applied Mathematics, vol. 30. SIAM, Philadelphia (2000)
Rao, S.C.S., Kumar, S., Kumar, M.: A parameter-uniform B-spline collocation method for singularly perturbed semilinear reaction-diffusion problem. J. Optim. Theory Appl. 146, 795–809 (2010)
Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. In: Springer Series in Computational Mathematics, 2nd edn., vol. 24. Springer, Berlin (2008)
Shishkin, G.I.: Grid approximation of singularly perturbed elliptic and parabolic equations. Second doctorial thesis. Keldysh Institute, Moscow (in Russian) (1990)
Surla, K., Uzelac, Z.: A spline difference scheme on a piecewise equidistant grid. Z. Angew. Math. Mech. 77(12), 901–909 (1997)
Vulanović, R.: A higher-order scheme for quasilinear boundary value problems with two small parameters. Computing 67(4), 287–303 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
H. Zarin is supported by the Ministry of Education and Science of the Republic of Serbia under grant 174030.
This publication has eminated from research conducted with support by the DAAD (grant no. 50740187) and the Ministry of Education and Science of the Republic of Serbia under grant “Collocation methods for singularly perturbed problems”.
Rights and permissions
About this article
Cite this article
Linß, T., Radojev, G. & Zarin, H. Approximation of singularly perturbed reaction-diffusion problems by quadratic C 1-splines. Numer Algor 61, 35–55 (2012). https://doi.org/10.1007/s11075-011-9529-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-011-9529-7
Keywords
- Reaction-diffusion problems
- Spline interpolation
- Spline collocation
- Singular perturbations
- A posteriori error estimation