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2023 | OriginalPaper | Chapter

Heptic Hermite Collocation on Finite Elements

Authors : Zanele Mkhize, Nabendra Parumasur, Pravin Singh

Published in: Frontiers in Industrial and Applied Mathematics

Publisher: Springer Nature Singapore

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Abstract

The chapter delves into the Heptic Hermite Collocation method on Finite Elements, a powerful tool for solving differential equations. It begins by introducing Orthogonal Collocation, its historical context, and its advantages over finite difference methods. The method's implementation is discussed, with a focus on its global and local applications. The chapter then explores the use of Hermite polynomials for basis functions, ensuring continuity and reducing the number of unknowns. Numerical examples are provided to illustrate the method's accuracy and efficiency, with a particular emphasis on its superior convergence rates and superconvergence properties. The chapter concludes with a detailed analysis of the method's performance on both linear and nonlinear boundary value problems, offering insights into its practical applications and theoretical foundations. This chapter is a must-read for anyone seeking to understand and apply advanced numerical methods in differential equations.

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Kantorovich L.V.: On an approximation method for the solution of a partial differential equation. Dokl. Akad. Nauk SSSR. 2, 532–536 (1934). arXiv:1904.04685

Frazer, R.A., Jones, W.P., Skan, S.W.: Approximation to Functions and to the Solutions of Differential Equations. Great Britain Aero. Res. Council, London, Report and Memo No. 1799 (1937)

Lanczos, C.: Trigonometric interpolation of empirical and analytical functions. J. Math. Phys. 17, 123–199 (1938). https://doi.org/10.1002/sapm1938171123

Slater, J.C.: Electronic energy bands in metal. Phys. Rev. 45, 794–801 (1934). http://orcid.org/10.1103/PhysRev.45.794

Runge, C.: Uber Empirische Functionen und die Interpolation Zwischen Aequidistanten Ordinaten. Zeitschrift fur Math. und Physik. 46, 224–243 (1901)MATH

Bert, C.W., Malik, M.: Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1–28 (1996). http://orcid.org/10.1115/1.3101882

Wright, K.: Chebyshev collocation methods for ordinary differential equations. Comp. J. 6, 358–365 (1964). http://orcid.org/10.1093/comjnl/6.4.358

Villadsen, J.: Selected Approximation Methods for Chemical Engineering Problems. Inst. for Kemiteknik Numer. Inst, Danmarks Tekniske Hojskole (1970)

Finlayson, B.A.: The Method of Weighted Residuals and Variational Principles. Academic Press, New York, NY (1972). http://orcid.org/10.1137/1.9781611973242

10.

Villadsen, J.V., Michelsen, M.L.: Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall, Englewood Cliffs, NJ (1978)MATH

11.

Michelsen, M.L., Villadsen, J.V.: A convenient computational procedure for collocation constants. Chem. Eng. J. 4, 64–68 (1972). http://orcid.org/10.1016/0300-9467(72)80054-6

12.

Michelsen, M.L., Villadsen, J.V.: Polynomial solution of differential equations. In: Mah, R.S.H., Seider, W.D. (eds.) Proceedings of an International Conference on Foundations of Computer-Aided Chemical Process Design, pp. 341–368 (1981)

13.

Finlayson, B.A.: Orthogonal collocation in chemical reaction engineering. Cat. Rev. Sci-Eng. 10, 69–138 (1974). http://orcid.org/10.1080/01614947408079627

14.

Finlayson, B.A.: Nonlinear Analysis in Chemical Engineering. Ravenna Park Publishing, Seattle (2003)

15.

Orzag, S.A.: Comparison of Pseudo Spectral and Spectral Approximation. Studies in Applied Mathematics, vol. 51, pp. 253–259 (1972). http://orcid.org/10.1002/sapm1972513253

16.

Gottlieb, D., Orzag, S.A.: Numerical analysis of spectral methods: theory and applications. SIAM, Philadelphia, PA (1977). http://orcid.org/10.1137/1.9781611970425

17.

Bellman, R., Casti, J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 134, 235–238 (1971)CrossRefMATH

18.

Bellman, R., Kashef, B.G., Casti, J.: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comp. Phys. 10, 40–52 (1972). https://doi.org/10.1016/0021-9991(72)90089-7

19.

Bellman, R.: Methods of Nonlinear Analysis, vol. 2. Academic Press, New York, (1973)

20.

Nielson, K.L.: Methods in Numerical Analysis. MacMillan, NY (1956)

21.

Zienkiewicz, O.C.: The Finite Element Method in Engineering Science. McGraw-Hill (1971)

22.

Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall (1973)

23.

Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall (1987)

24.

Sharma, S., Jiwari, R., Kumar, S.: Numerical solutions of two point boundary value problems using Galerkin-Finite element method. Int J Nonlinear Sci. 13(2), 204–210 (2012)MATH

25.

Sharma, D., Jiwari, R., Kumar, S.: A comparative study of Modal matrix and finite elements methods for two point boundary value problems. Int. J. Appl. Math. Mech. 8(13), 29–45 (2012)

26.

Yadav, O.P., Jiwari, R.: Finite element approach to capture Turing patterns of autocatalytic Brusselator model. J. Math. Chem. 57(3), 769–789 (2019)CrossRefMATH

27.

Yadav, O.P., Jiwari, R.: Finite element approach for analysis and computational modelling of coupled reaction diffusion models. Numer. Methods Partial Differ. Equ. 35(2), 830–850 (2019)CrossRefMATH

28.

de Boor C., Swartz B.: Collocation at gaussian points. In: Society for Industrial and Applied Mathematics, vol. 10, pp. 582–606 (1973). http://orcid.org/10.1137/0710052. (SIAM J. Numer. Anal.)

29.

Douglas Jr., J., Dupont, T.: A finite element collocation method for quasilinear parabolic equations. Math. Comp. 27, 17–28 (1973). http://orcid.org/10.2307/2005243

30.

Carey, G., Finlayson, B.A.: Orthogonal collocation on finite elements. Chem. Eng. Sci. 30, 587–596 (1975). http://orcid.org/10.1016/0009-2509(75)80031-5

31.

Diaz, J.: A hybrid collocation-Galerkin method for two-point boundary value problems using continuous piecewise polynomial spaces. Ph.D. Thesis, Rice University (1975). https://hdl.handle.net/1911/15125

32.

Dunn, R., Wheeler, M.F.: Some collocation-Galerkin methods for two-point boundary value problems. SIAM J. Numer. Anal. 13(5), 720–733 (1976)CrossRefMATH

33.

Wheeler, M.F.: A \(C^0\)-collocation-finite element method for two-point boundary value and one space dimension parabolic problems. SIAM J. Numer. Anal. 14(1), 71–90 (1977). http://orcid.org/10.1137/0714005

34.

Gray, W.G.: An Efficient Finite Element Scheme for Two-Dimensional Surface Water Computations. Finite Elements in Water Resources. In: Gray, W.G., Pinder, G.F., Brebbia, C.A. (eds.). Pentech Press, London (1977)

35.

Young, L.C.: A preliminary comparison of finite element methods for reservoir simulation. In: Vichnevetsky, R. (ed.) Advances in Computer Methods for Partial Differential Equations-II. IMACS(AICA). vol. 2, pp. 307–320. Rutgers U., New Brunswick, N.J. (1977)

36.

Young, L.C.: A finite-element method for reservoir simulation. Soc. Petr. Eng. J. 21(1), 115–128 (1981). http://orcid.org/10.2118/7413-PA

37.

Hennart, J.P.: Topics in Finite Element Discretization of Parabolic Evolution Problems. Lecture Notes in Math, vol. 909. Springer, Berlin, Heidelberg (1982)

38.

Leyk, Z.: A \(C^0\)-collocation-like method for boundary value problems. Numerische Mathametik, 49, 39–54 (1986). http://eudml.org/doc/133097

39.

Leyk, Z.: A \(C^0\)-collocation-like method for elliptic equations on rectangular regions. J. Austral. Math. Soc. Ser. B 38, 368–387 (1997). http://orcid.org/10.1017/S0334270000000734

40.

Maday, Y., Patera, A.T.: Spectral element methods for the incompressible Navier-Stokes equations. In: Noor, A.K. (ed.) State-of-the-Art Surveys on Computational Mechanics. ASME, New York (1989)

41.

Canuto, C., Hussaini, M., Quarteroni, A., Zang, T., Jr.: Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)CrossRefMATH

42.

Karniadakis, G., Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics: Second Edition (Numerical Mathematics and Scientific Computation). Oxford University Press (2013)

43.

Vosse, van de, F.N., Minev, P.D.: Spectral elements methods: theory and applications. EUT Report 96-W-001, Eindhoven University of Technology (1996)

Title: Heptic Hermite Collocation on Finite Elements
Authors: Zanele Mkhize
Nabendra Parumasur
Pravin Singh
Publisher: Springer Nature Singapore
Book: Frontiers in Industrial and Applied Mathematics
Print ISBN: 978-981-19-7271-3

Electronic ISBN: 978-981-19-7272-0

Copyright Year: 2023
DOI: https://doi.org/10.1007/978-981-19-7272-0_38

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