A Jacobi spectral galerkin method for the integrated forms of fourth-order elliptic differential equations

Numerical Methods for Partial Differential Equations

Volumn 25, Issue 3, 2009, Pages 712-739

  Doha, Eid H ^a   Bhrawy, Ali H ^b  

^a CAIRO UNIVERSITY   (Egypt)

^b Beni Suef University   (Egypt)

Author keywords

Biharmonic operator; Direct solver; Fourth order differential equations; Jacobi polynomials; Spectral method

Indexed keywords

EID: 66049087571     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20369     Document Type: Article

References (51)

1. 1 J. P. Boyd, Chebyshev and Fourier Spectral methods, 2nd Ed., Dover Publications, Mineola, 2001.
2. 2 E. A. Coutsias, T. Hagstrom, and D. Torres, An efficient spectral method for ordinary differential equations with rational function, Math Comp 65 (1996), 611–635.
3. 3 D. Funaro, Polynomial approximation of differential equations, Lecturer Notes in Physics, Springer‐Verlag, Heidelberg, Berlin, New York, 1992.
4. 4 D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods: theory and applications, SIAM, Philadelphia, 1977.
5. 5 G. Szegö, Orthogonal polynomials, Am Math Soc Colloq Pub 23 (1985).
6. 6 C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer‐Verlag, New York, 1989.
7. 7 R. G. Voigt, D. Gottlieb, and M. Y. Hussaini, Spectral methods for partial differential equations, SIAM, Philadelphia, 1984.
8. 8 T. N. Phillips and A. Karageorghis, On the coefficients of integrated expansions of ultraspherical polynomials, SIAM J Numer Anal 27 (1990), 823–830.
9. 9 E. H. Doha, On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations, J Comput Appl Math 139 (2002), 275–298.
10. 10 E. H. Doha, Explicit formulae for the coefficients of integrated expansions for Jacobi polynomials and their integrals, Integral Transforms Special Functions 14 (2003), 69–86.
11. 11 B. Bialecki and A. Karageorghis, A Legendre spectral collocation method for the biharmonic dirichlet problem, Math Model Numer Anal 34 (2000), 637–662.
12. 12 B. Bialecki and A. Karageorghis, A Legendre spectral Galerkin method for the biharmonic Dirichlet problem, SIAM J Sci Comput 22 (2000), 1549–1569.
13. 13 P. Bjørstad, Fast numerical solution of the biharmonic Dirichlet problem on rectangles, SIAM J Numer Anal 20 (1983), 59–71.
14. 14 P. E. Bjørstad and B. P. Tjøstheim, Efficient algorithms for solving a fourth‐order equation with the spectral‐Galerkin method, SIAM J Sci Comput 18 (1997), 621–632.
15. 15 B.‐Y. Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J Math Anal Appl 243 (2000), 373–408.
16. 16 B.‐Y. Guo and L.‐I. Wang, Jacobi interpolation approximations and their applications to singular differential equations, Adv Comput Math 14 (2001), 227–276.
17. 17 I. Babuss̃ka and B. Q. Guo, Optimal estimates for lower and upper bounds of approximation errors in the p‐version of the finite element method in two‐dimensions, Numer Math 85 (2000), 219–255.
18. 18 I. Babuss̃ka and B. Q. Guo, Direct and inverse approximation theorems for the p‐version of the finite element method in the framework of weighted Besov spaces, I. Approximability of functions in the weighted Besov spaces, SIAM J Numer Anal 39 (2002), 1512–1538.
19. 19 B.‐Y. Guo, J. Shen, and Z.‐Q. Wang, Chebyshev rational spectral and pseudospectral methods on a semi‐infinite interval, Internat J Numer Methods Eng 53 (2002), 65–84.
20. 20 E. P. Stephan and M. Suri, On the convergence of the p‐version of the boundary element Galerkin method, Math Comp 52 (1989), 31–48.
21. 21 Z.‐Q. Wang and B.‐Y. Guo, A rational approximation and its applications to nonlinear partial differential equations on the whole line, J Math Anal Appl 274 (2002), 374–403.
22. 22 W. Heinrichs, Improved condition number of spectral methods, Math Comp 53 (1989), 103–119.
23. 23 J. Shen, Efficient spectral‐Galerkin method I: Direct solvers of second‐and fourth‐order equations using Legendre polynomials, SIAM J Sci Comput 15 (1994), 1489–1505.
24. 24 J. Shen, Efficient spectral‐Galerkin method II: Direct solvers of second‐and fourth‐order equations using Chebyshev polynomials, SIAM J Sci Comput 16 (1995), 74–87.
25. 25 E. H. Doha and A. H. Bhrawy, Efficient spectral‐Galerkin algorithms for direct solution of the integrated forms for second‐order equations using ultraspherical polynomials, ANZIAM J 48 (2007), 361–386.
26. 26 M. Fernandino, C. A. Dorao, and H. A. Jakobsen, Jacobi Galerkin spectral method for cylindrical and spherical geometries, Chem Eng Sci 62 (2007), 6777–6783.
27. 27 G. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, Part I: Derivation of basic equations, Acta Astronaut 4 (1977), 1177–1206.
28. 28 B. L. Buzbee and F. W. Dorr, The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular domains, SIAM J Numer Anal 8 (1971), 722–736.
29. 29 W. Heinrichs, Line relaxation for spectral multigrid methods, J Comput Phys 77 (1988), 166–182.
30. 30 W. Heinrichs, Collocation and full multigrid methods, Appl Math Comput 26 (1988), 35–43.
31. 31 T. A. Zang, Y. S. Wong, and M. Y. Hussaini, Spectral multigrid methods for elliptic equations I, J Comput Phys 48 (1982), 485–501.
32. 32 T. A. Zang, Y. S. Wong, and M. Y. Hussaini, Spectral mutigrid methods for elliptic equations II, J Comput Phys 54 (1984), 489–507.
33. 33 C. Bernardi and Y. Maday, Some spectral approximations of fourth‐order problems, P. Nevai, A. Pinkus, editors, Progress in approximation theory, Academic Press, New York, 1991.
34. 34 D. Funaro and W. Heinrichs, Some results about the pseudospectral approximation of one dimensional fourth‐order problems, Numer Math 58 (1990), 399–418.
35. 35 C. Bernardi and Y. Maday, Spectral methods for the approximation of fourth‐order problems: applications to the stokes and Navier‐Stokes equations, Comput Struct 30 (1988), 205–216.
36. 36 C. Bernardi and Y. Maday, Some spectral approximations of one‐dimensional fourth‐order problems, ICASE, NASA‐Langley Research Center, Hampton, VA, 1989.
37. 37 A. Karageorghis and T. N. Phillips, Efficient direct methods for solving the spectral collocation equations for Stokes flow in rectangularly decomposable domains, SIAM J Sci Statist Comput 10 (1989), 89–103.
38. 38 A. Karageorghis and T. N. Phillips, Spectral collocation methods for Stokes flow in contraction geometries and unbounded domains, J Comput Phys 80 (1989), 314–330.
39. 39 A. H. Bhrawy, Spectral galerkin method for solving the integrated forms of second‐ and fourth‐order differential equations by using ultraspherical polynomials, Cairo University, Egypt, M.Sc. Thesis, 2003.
40. 40 Y. Luke, The special functions and their approximations, vol. 2, Academic Press, New York, 1969.
41. 41 E. H. Doha, On the coefficients of differentiated expansions and derivatives of Jacobi polynomials, J Phys A Math Gen 35 (2002), 3467–78.
42. 42 G. N. Watson, A note on generalized hypergeometric series, Proc London Math Soc, (2) 23 xiii ‐ xv (1925), (Records for 8 Nov. 1923).
43. 43 E. H. Doha, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J Phys A Math Gen 37 (2004), 657–675.
44. 44 E. H. Doha, The ultrasherical coefficients of the moments of a general‐order derivative of an infinitely differentiable function, J Comput Appl Math 89 (1998), 53–72.
45. 45 Y. Luke, Mathematical functions and their approximations, Academic Press, New York, 1975.
46. 46 E. H. Doha and A. H. Bhrawy, Efficient spectral‐Galerkin algorithms for direct solution for second‐order differential equations using Jacobi polynomials, Numer Algorith 42 (2006), 137–164.
47. 47 E. H. Doha and A. H. Bhrawy, Efficient Spectral‐Galerkin algorithms for direct solution of fourth‐order differential equations using Jacobi polynomials, Appl Numer Math (2007), doi:10.1016/j.apnum.2007.07.001.
48. 48 A. Graham, Kronecker products and matrix calculus: with applications, Ellis Horwood, England, 1981.
49. 49 A. Malek and T. N. Phillips, Pseudospectral collocation methods for fourth order differential equations, IMA J Numer Anal 15 (1995), 523–553.
50. 50 B. K. Alperto and V. Rokhlin, A fast algorithm for the evaluation of Legendre expansions, SIAM J Sci Stat Comput 12 (1991), 158–179.
51. 51 J. Shen, Efficient Chebyshev Legendre Galerkin methods for elliptic problems, Houston J Math (1996), 233–240.