Collocation Method to Solve Elliptic Equations, Bivariate Poly-Sinc Approximation
Keywords:
Elliptic PDEs; Bivariate approximation; Lagrange interpolation; Sinc points; Conformal maps; Poly-Sinc methods; Collocation method; Dirichlet boundary conditions; Mixed boundary conditions.
Abstract
The paper proposes a collocation method to solve bivariate elliptic partial differential equations. The method uses Lagrange approximation based on Sinc point collocations. The proposed approximation is collocating on non-equidistant interpolation points generated by conformal maps, called Sinc points. We prove the upper bound of the error for the bivariate Lagrange approximation at these Sinc points. Then we define a collocation algorithm using this approximation to solve elliptic PDEs. We verify the Poly-Sinc technique for different elliptic equations and compare the approximate solutions with exact solutions.
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References
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[14] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations Steady State and Time Dependent Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Soft cover / ISBN978-0-898716-29-0 xiv+339, (2007).
[15] S.J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math.Inform., 33, pp. 109-123, (2006). [16] P. Erdös, Problems and results on the theory of interpolation. II, ActaMath. Hungar., 12, pp. 235-244, (1961).
[17] B. Bialecki, Sinc-Collocation Methods for Two-Point Boundary Value Problems, IMA J. Num. Anal., 11, pp. 357-375, (1991).
[18] F. Stenger, H. A. El-Sharkawy and, G. Baumann, The Lebesgue Constantfor Sinc Approximations, New Perspectives on Approximation and Sampling Theory - Festschrift in the honor of Paul Butzer's 85th birthday. Eds. A.Zayed and G. Schmeisser, Birkhaeuser, Busel, pp. 319-335, (2014).
[19] M. Youssef, H.A. El-Sharkawy and, G. Baumann Lebesgue Constants for Lagrange Interpolation at Sinc-Data, to be published.
[20] M. A. Golberg, C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, [In:] Boundary Integral Methods, Numerical and Mathematical Aspects, Golberg M.A. (Edit.), Boston, Computational Mechanics Publications, pp. 103-176, (1998).
[21] N. R. Aluru A point collocation method based on reproducing kernel approximations, Int. J. Numer. Math. Eng., 47, pp. 1083-1121, (2000).
[22] N.J. Lybeck, Domain Decomposition Methods for Elliptic Problems, Ph.D.Thesis, Montana State University, (1994).
[2] D. R. Lynch, Numerical Partial Differential Equations for Environmental Scientists and Engineers, A First Practical Course, Springer, (2005).
[3] J. D. Logan, Applied Partial Differential Equations, 3rd edition, Springer International Publishing Switzerland, (2015).
[4] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and QuasiLinear Elliptic Equations, English version, Academic Press New York and London, 46,(1968). [5] C. Runge, Über empirische Funktionen und die Interpolation zwischenaquidistanten Ordinaten, Zeitschriftfür Mathematik und Physik 46, pp. 224-243, (1901).
[6] F. Stenger, M. Youssef, and J. Niebsch, Improved Approximation via Use of Transformations: In: Multiscale Signal Analysis and Modeling, Eds. X.Shen and A.I. Zayed, NewYork: Springer, pp. 25-49, (2013).
[7] F. Stenger, Handbook of Sinc Methods, CRC Press , (2011).
[8] L.N. Trefethen, Spectral Methods in MATLAB, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2000).
[9] I. Babuska, W.C. Rheinboldt, Error estimates for adaptive finite elementcomputations, SIAM J. Numer. Anal. 15 pp. 736-754, (1978).
[10] I. Babuska, A. Miller, A feedback finite element method with a posteriorierror estimation. part i: The finite element method and some basic properties of the a posteriori error estimates, Comp. Math. Appl. Mech. Eng. 61 (1987)1-40. O.V. Vasilyev, N.K.-R. Kevlahan / Journal of Computational Physics206, pp. 412-431, (2005).
[11] R.E. Bank, A.Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comput. 44 (170), pp. 283-301, (1985).
[12] F.A. Bornemann, B. Erdmann, R. Kornhuber, A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer.Anal. 33 (3), pp. 1188-1204, (1996).
[13] S. Dahlke, W. Dahmen, R. Hochmuth, R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math. 23, pp.21-47, (1997).
[14] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations Steady State and Time Dependent Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Soft cover / ISBN978-0-898716-29-0 xiv+339, (2007).
[15] S.J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math.Inform., 33, pp. 109-123, (2006). [16] P. Erdös, Problems and results on the theory of interpolation. II, ActaMath. Hungar., 12, pp. 235-244, (1961).
[17] B. Bialecki, Sinc-Collocation Methods for Two-Point Boundary Value Problems, IMA J. Num. Anal., 11, pp. 357-375, (1991).
[18] F. Stenger, H. A. El-Sharkawy and, G. Baumann, The Lebesgue Constantfor Sinc Approximations, New Perspectives on Approximation and Sampling Theory - Festschrift in the honor of Paul Butzer's 85th birthday. Eds. A.Zayed and G. Schmeisser, Birkhaeuser, Busel, pp. 319-335, (2014).
[19] M. Youssef, H.A. El-Sharkawy and, G. Baumann Lebesgue Constants for Lagrange Interpolation at Sinc-Data, to be published.
[20] M. A. Golberg, C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, [In:] Boundary Integral Methods, Numerical and Mathematical Aspects, Golberg M.A. (Edit.), Boston, Computational Mechanics Publications, pp. 103-176, (1998).
[21] N. R. Aluru A point collocation method based on reproducing kernel approximations, Int. J. Numer. Math. Eng., 47, pp. 1083-1121, (2000).
[22] N.J. Lybeck, Domain Decomposition Methods for Elliptic Problems, Ph.D.Thesis, Montana State University, (1994).
Published
2016-05-16
How to Cite
Youssef, M., & Baumann, G. (2016). Collocation Method to Solve Elliptic Equations, Bivariate Poly-Sinc Approximation. Journal of Progressive Research in Mathematics, 7(3), 1079-1091. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/744
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