Abstract
Spectral methods are approximation techniques for the computation of the solutions to ordinary and partial differential equations. They are based on a polynomial expansion of the solution. The precision of these methods is limited only by the regularity of the solution, in contrast to the finite difference method and the finite element methods. The approximation is based primarily on the variational formulation of the continuous problem. The test functions are polynomials and the integrals involved in the formulation are computed by suitable quadrature formulas. This project proposes to implement a spectral method to solve the following boundary value problem defined on the interval Ω = (−1, 1):
with f ∈ L 2 (Ω) and c a positive real number.
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References
C. Bernardi, M. Dauge, and Y. Maday, Spectral methods for axisymmetric domains. Numerical algorithms and tests due to Mejdi Azaiez, Series in Applied Mathematics, 3. Gauthier-Villars, North-Holland, Amsterdam, 1999.
C. Bernardi and Y. Maday, Spectral Methods in Handbook of numerical analysis, Vol. V, North-Holland, Amsterdam, 1997.
C. Bernardi, Y. Maday, and F. Rapetti, Discrétisations variationnelles de problémes aux limites elliptiques, Mathématiques & Applications, Vol. 45, Springer-Verlag, Mai 2004.
P. J. Davis, Interpolation and Approximation, Dover Publications, Inc., New York, 1975.
A. R. Krommer and C. W. Ueberhuber, Numerical Integration on Advanced Computer Systems, Lecture Notes in Computer Science, 848, Springer-Verlag, Berlin, 1994.
G. Szegő, Orthogonal Polynomials, fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII, Providence, R.I., 1975.
B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lecture Notes in Computational Science and Engineering, 17, Springer-Verlag, Berlin, 2001.
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(2007). Solving a Differential Equation by a Spectral Method. In: Danaila, I., Joly, P., Kaber, S.M., Postel, M. (eds) An Introduction to Scientific Computing. Springer, New York, NY. https://doi.org/10.1007/978-0-387-49159-2_5
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DOI: https://doi.org/10.1007/978-0-387-49159-2_5
Publisher Name: Springer, New York, NY
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