Abstract
We construct a new numerical method comprising upwind finite difference operators on asymptotically appropriate Shishkin meshes to obtain a numerical approximation to the solution of the Hemker problem. Numerical results indicate that the numerical approximations are computationally uniformly convergent with respect to the small parameter ε.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Let \(U_\varepsilon ^N\) be the computed solutions on certain meshes \(\varOmega _\varepsilon ^N\). Define the maximum two-mesh global differences by
$$\displaystyle \begin{aligned} D^N_\varepsilon:= \Vert \bar U_\varepsilon^N-\bar U_\varepsilon^{2N}\Vert_{\varOmega_\varepsilon^N \cup \varOmega_\varepsilon^{2N}} \quad \text{and} \quad D^N:= \max_{\varepsilon} D^N_\varepsilon, \end{aligned} $$(2.3a)where \(\bar U^N\) denotes the bilinear interpolation of the discrete solution U N on the mesh \(\varOmega _\varepsilon ^{N}\). Then, for any particular value of ε and N, the computed orders of convergence are \(\bar p^{N}_\varepsilon \) and, for any particular value of N and all values of ε, the parameter-uniform computed orders of convergence \(\bar p^N\) are defined, respectively, by
$$\displaystyle \begin{aligned} \bar p^N_\varepsilon:= \log_2\left (\frac{D^N_\varepsilon}{D^{2N}_\varepsilon} \right) \quad \text{and} \quad \bar p^N := \log_2\left (\frac{D^N}{D^{2N}} \right). \end{aligned} $$(2.3b)
References
Augustin, M., Caiazzo, A., Fiebach, A., Fuhrmann, J., John, V., Linke, A., Umla, R.: An assessment of discretizations for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Eng. 200(47–48), 3395–3409 (2011)
Farrell, P.A., Hegarty, A.F.: On the determination of the order of uniform convergence. In: Proceedings of the 13th IMACS World Congress, Dublin, pp. 501–502 (1991)
Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman and Hall/CRC Press, Boca Raton (2000)
Han, H., Huang, Z., Kellogg, R.B.: A tailored finite point method for a singular perturbation problem on an unbounded domain. J. Sci. Comput. 36(2), 243–261 (2008)
Hegarty, A.F., O’Riordan, E.: Numerical results for singularly perturbed convection-diffusion problems on an annulus. In: Huang, Z., Stynes, M., Zhang, Z. (eds.) Proceedings of the International Conference on Boundary and Interior Layers - Computational and Asymptotic Methods, BAIL 2016, Beijing, August 2016. Lecture Notes in Computational Science and Engineering, vol. 120, pp. 101–112. Springer, New York (2017)
Hegarty, A.F. O’Riordan, E.: A parameter-uniform numerical method for a singularly perturbed convection-diffusion problem posed on an annulus. Comput. Math. Appl. 78(10), 3329–3344 (2019)
Hemker, P.W.: A singularly perturbed model problem for numerical computation. J. Comp. Appl. Math. 76, 277–285 (1996)
Il’in, A.M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Mathematical Monographs, vol. 102. American Mathematical Society, Providence (1992)
Lagerstrom, P.A.: Matched Asymptotic Expansions: Ideas and Techniques. Applied Mathematical Sciences, vol. 76. Springer, New York (1988)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Hegarty, A.F., O’Riordan, E. (2020). A Numerical Method for the Hemker Problem. In: Barrenechea, G., Mackenzie, J. (eds) Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2018. Lecture Notes in Computational Science and Engineering, vol 135. Springer, Cham. https://doi.org/10.1007/978-3-030-41800-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-41800-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-41799-4
Online ISBN: 978-3-030-41800-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)