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01.09.2018

Analysis of the SDFEM for singularly perturbed differential–difference equations

verfasst von: Li-Bin Liu, Haitao Leng, Guangqing Long

Erschienen in: Calcolo | Ausgabe 3/2018

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In this paper, the stability and accuracy of a streamline diffusion finite element method (SDFEM) for the singularly perturbed differential–difference equation of convection term with a small shift is considered. With a special choice of the stabilization quadratic bubble function and by using the discrete Green’s function, the new method is shown to have an optimal second order in the sense that \(\Vert u-u_{h}\Vert _{\infty }\le C\inf \nolimits _{v_h\in V^h}\Vert u-v_{h}\Vert _{\infty }\), where \(u_{h}\) is the SDFEM approximation of the exact solution u in linear finite element space \(V_{h}\). At last, a second order uniform convergence result for the SDFEM is obtained. Numerical results are given to confirm the \(\varepsilon \)-uniform convergence rate of the nodal errors.

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Longtin, A., Milton, J.: Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Math. Biosci. 90, 183–199 (1988)MathSciNetCrossRef

Mackey, M.C., Glass, L.: Oscillations and chaos in physiological control systems. Science 197, 287–289 (1977)CrossRef

Wazewska-Czyzewska, M., Lasota, A.: Mathematical models of the red cell system. Mat. Stos. 6, 25–40 (1976)

Derstine, M.W., Gibbs, H.M., Kaplan, D.L.: Bifurcation gap in a hybrid optical system. Phys. Rev. A 26, 3720–3722 (1982)CrossRef

Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary-value problems for differential–difference equation. SIAM J. Appl. Math. 42, 502–531 (1982)MathSciNetCrossRef

Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary-value problems for differential–difference equations. V. Small shifts with layer behavior. SIAM J. Appl. Math. 54, 249–272 (1994)MathSciNetCrossRef

Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary-value problems for differential–difference equations. VI. Small shifts with rapid oscillations. SIAM J. Appl. Math. 54, 273–283 (1994)MathSciNetCrossRef

Amiraliyeva, I.G., Erdogan, F., Amiraliyev, G.M.: A uniform numerical method for dealing with a singularly perturbed delay initial value problem. Appl. Math. Lett. 23, 1221–1225 (2010)MathSciNetCrossRef

Zarin, H.: On discontinuous Galerkin finite element method for singularly perturbed delay differential equations. Appl. Math. Lett. 38, 27–32 (2014)MathSciNetCrossRef

10.

Avudai Selvi, P., Ramanujam, N.: An iterative numerical method for singularly perturbed reaction–diffusion equations with negative shift. J. Comput. Appl. Math. 296, 10–23 (2016)MathSciNetCrossRef

11.

Erdogan, F., Cen, Z.: A uniformly almost second-order convergent numerical method for singularly perturbed delay differential equations. J. Comput. Appl. Math. 333, 382–394 (2018)MathSciNetCrossRef

12.

Subburayan, V., Mahendran, : An \(\varepsilon \)-uniform numerical method for third singularly perturbed delay differential equations with discontinuous convection coefficient and source term. Appl. Math. Comput. 331, 404–415 (2018)MathSciNet

13.

Geng, F.Z., Qian, S.P.: Modified reproducing kernel method for singularly perturbed boundary value problems with a delay. Appl. Math. Model. 39, 5592–5597 (2015)MathSciNetCrossRef

14.

Kadalbajoo, M.K., Ramesh, V.P.: Hybrid method for numerical solution of singulary perturbed delay differential equations. Appl. Math. Comput. 187, 797–814 (2007)MathSciNetMATH

15.

Kadalbajoo, M.K., Ramesh, V.P.: Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptaion strategy. Appl. Math. Comput. 188, 1816–1831 (2007)MathSciNetMATH

16.

Patidar, K.C., Sharma, K.K.: \(\varepsilon \)-Uniformly convergent non-stand finite difference methods for singularly perturbed differential difference equations with small delay. Appl. Math. Comput. 175, 864–890 (2006)MathSciNetMATH

17.

Kadalbajoo, M.K., Kumar, D.: Fitted mesh \(B\)-spline collocation method for singularly perturbed differential–difference equations with small delay. Appl. Math. Comput. 204, 90–98 (2008)MathSciNetMATH

18.

Kadalbajoo, M.K., Kumar, D.: A computational method for singularly perturbed nonlinear differential–difference equations with small shift. Appl. Math. Model. 34, 2584–2596 (2010)MathSciNetCrossRef

19.

Rao, R.N., Chakravarthy, R.P.: A finite difference method for singularly perturbed differential–difference equations with layer and oscillatory behavior. Appl. Math. Model. 37, 5743–5755 (2013)MathSciNetCrossRef

20.

Geng, F.Z., Qian, S.P., Cui, M.G.: Improved reproducing kernel method for singularly perturbed differential–difference equations with boundary layer behavior. Appl. Math. Comput. 252, 58–63 (2015)MathSciNetMATH

21.

Adivi Sri Venkata, R.K., Palli, M.M.K.: A numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method. Calcolo (2017). https://doi.org/10.1007/s10092-017-0215-6 MathSciNetCrossRef

22.

Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (ed.) Finite Element Methods for Convection Dominated Flows, AMD, vol. 34. ASME, New York (1979)

23.

Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comput. 66, 31–44 (1997)MathSciNetCrossRef

24.

Linß, T., Stynes, M.: The SDFEM on Shishkin meshes for linear convection–diffusion problem. Numer. Math. 87, 457–484 (2001)MathSciNetCrossRef

25.

Roos, H.G., Zarin, H.: The streamline-diffusion method for a convection–diffusion problem with a point source. J. Comput. Appl. Math. 150, 109–128 (2003)MathSciNetCrossRef

26.

Stynes, M., Tobiska, L.: The SDFEM for a convection–diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal. 41, 1620–1642 (2003)MathSciNetCrossRef

27.

Sangalli, G.: Quasi optimality of the supg method for the one-dimensional advection–diffusion problem. SIAM J. Numer. Anal. 41, 1528–1542 (2003)MathSciNetCrossRef

28.

Chen, L., Xu, J.: Stability and accuracy of adapted finite element methods for singularly perturbed problems. Numer. Math. 109, 167–191 (2008)MathSciNetCrossRef

29.

Chen, L., Xu, J.: An optimal streamline diffusion finite element method for a singularly perturbed problem. In: AMS Contemporary Mathematics Series: Rencent Advances in Adaptive Computation, vol. 383, pp. 236–246, HangZhou (2005)

30.

Chen, L., Wang, Y., Xu, J.: Stability of a streamline diffusion finite element method for turning point problems. J. Comput. Appl. Math. 220, 712–724 (2008)MathSciNetCrossRef

31.

Celiker, F., Zhang, Z., Zhu, H.: Nodal superconvergence of SDFEM for singularly perturbed problems. J. Sci. Comput. 50, 405–433 (2012)MathSciNetCrossRef

32.

Zhang, J., Liu, X.: Supercloseness of the SDFEM on Shishkin triangular meshes for problems with exponential layer. Adv. Comput. Math. 43, 759–775 (2017)MathSciNetCrossRef

33.

Liu, X., Zhang, J.: Analysis of the SDFEM in a streamline diffusion norm for singularly perturbed convection diffusion problems. Appl. Math. Lett. 69, 61–66 (2017)MathSciNetCrossRef

34.

Zhang, J., Liu, X.: Pointwise estimates of SDFEM on Shishkin triangular meshes for problems with exponential layers. BIT Math. Numer. (2017). https://doi.org/10.1007/s10543-017-0661-1 MathSciNetCrossRef

35.

Sahihi, H., Abbasbandy, S., Allahviranloo, T.: Reproducing kernel method for solving singularly perturbed differential–difference equations with boundary layer behavior in Hilbert space. J. Comput. Appl. Math. 328, 30–43 (2018)MathSciNetCrossRef

36.

37.

Venkata, R.K.A.S., Palli, M.M.K.: A numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method. Calcolo (2017). https://doi.org/10.1007/s10092-017-0215-6 MathSciNetCrossRef

38.

Mohapatra, J., Natesan, S.: Uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid. Numer. Math. Theory. Methods Appl. 3, 1–22 (2010)MathSciNetMATH

39.

Andreev, V.B.: The green function and a priori estimates of solutions of monotone three-point singularly perturbed finite-difference schemes. Differ. Equ. 37, 923–933 (2001)MathSciNetCrossRef

40.

Linß, T.: Layer-adapted meshes for convectioni-diffusion problems. Comput. Methods Appl. Mech. Eng. 192, 1061–105 (2003)MathSciNetCrossRef

41.

Qiu, Y., Sloan, D., Tang, T.: Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: analysis of convergence. J. Comput. Appl. Math. 116, 121–143 (2000)MathSciNetCrossRef

42.

Chen, Y.: Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution. J. Comput. Appl. Math. 159, 25–34 (2003)MathSciNetCrossRef

43.

Chen, Y.: Uniform convergence analysis of finite difference approximations for singular perturbation problems. Adv. Comput. Math. 24, 197–212 (2006)MathSciNetCrossRef

44.

Kopteva, N., Stynes, M.: A robust adaptive method for quasi-linear one-dimensional convection diffusion problem. SIAM J. Numer. Anal. 39, 1446–1467 (2001)MathSciNetCrossRef

Titel: Analysis of the SDFEM for singularly perturbed differential–difference equations
verfasst von: Li-Bin Liu
Haitao Leng
Guangqing Long
Publikationsdatum: 01.09.2018
Verlag: Springer International Publishing
Erschienen in: Calcolo / Ausgabe 3/2018
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-018-0265-4

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