Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Coupling finite element and spectral methods: first results


Authors: Christine Bernardi, Naïma Debit and Yvon Maday
Journal: Math. Comp. 54 (1990), 21-39
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1990-0995205-7
MathSciNet review: 995205
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A Poisson equation on a rectangular domain is solved by coupling two methods: the domain is divided in two squares; a finite element approximation is used on the first square and a spectral discretization is used on the second. Two kinds of matching conditions on the interface are presented and compared; in both cases, error estimates are proved.


References [Enhancements On Off] (What's this?)

  • [1] I. Babuška, B. Szabo, and I. N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal. 18 (1981), 515-545. MR 615529 (82j:65081)
  • [2] J. Berg and J. Löfström, Interpolation spaces: An introduction, Springer-Verlag, Berlin, Heidelberg, and New York, 1976. MR 0482275 (58:2349)
  • [3] C. Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal. 26 (1989), 1212-1240. MR 1014883 (91a:65228)
  • [4] -, Interpolation par éléments finis de fonctions tout juste continues et application (in preparation).
  • [5] C. Bernardi, N. Debit, and Y. Maday, Couplage de méthodes spectrale et d'éléments finis: premiers résultats d'approximation, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 353-356. MR 909564 (88h:65200)
  • [6] C. Bernardi and Y. Maday, Some spectral approximations of one-dimensional fourth-order problems, J. Approx. Theory (to appear). MR 1114769 (92j:65176)
  • [7] P. E. Bjørstad and O. B. Widlund, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal. 23 (1986), 1097-1020. MR 865945 (88h:65188)
  • [8] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods with applications to fluid dynamics, Springer-Verlag, Berlin, Heidelberg, and New York, 1987. MR 2223552 (2007c:65001)
  • [9] C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), 67-86. MR 637287 (82m:41003)
  • [10] -, Spectral and pseudo-spectral methods for parabolic problems with nonperiodic boundary conditions, Calcolo 18 (1981), 197-218. MR 647825 (84h:35132)
  • [11] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, New York, and Oxford, 1978. MR 0520174 (58:25001)
  • [12] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, RAIRO 7-R3 (1973), 33-75. MR 0343661 (49:8401)
  • [13] P. J. Davis and P. Rabinowitz, Methods of numerical integration, Academic Press, Orlando, 1985. MR 760629 (86d:65004)
  • [14] N. Debit, Thesis (in preparation).
  • [15] D. Funaro, A multidomain spectral approximation of elliptic equations, Numer. Methods Partial Differential Equations 2 (1986), 187-205. MR 925372 (88m:65183)
  • [16] D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods, SIAM CBMS, Philadelphia, 1977.
  • [17] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, Boston, London, and Melbourne, 1985. MR 775683 (86m:35044)
  • [18] Handbook of mathematical functions (M. Abramowitz and I. A. Stegun, eds.), Dover, New York, 1970.
  • [19] K. Z. Korczak and A. T. Patera, An isoparametric spectral element method and its application to incompressible two-dimensional flow in complex geometries, Proc. 6th GAMM Conference on Numerical Methods in Fluid Mechanics, Vieweg, 1986.
  • [20] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vols. I & II, Dunod, Paris, 1968.
  • [21] Y. Maday, Analysis of spectral projectors in one-dimensional domains, Math. Comp. (to appear). MR 1035939 (91c:41095)
  • [22] -, Analysis of spectral projectors in multi-dimensional domains, Math. Comp. (to appear).
  • [23] Y. Maday and A. T. Patera, Spectral element methods for the incompressible Navier-Stokes equations, State of the Art Surveys in Computational Mechanics (A. Noor and J. T. Oden, eds.), A.S.M.E., 1989, pp. 71-143.
  • [24] Y. Maday and A. Quarteroni, Legendre and Chebyshev spectral approximations of Burgers' equation, Numer. Math 37 (1981), 321-332. MR 627106 (83c:65246)
  • [25] A. T. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys. 54 (1984), 468-488.
  • [26] A. Quarteroni, Some results of Bernstein and Jackson type for polynomial approximation in $ {{\mathbf{L}}^p}$-spaces, Japan J. Appl. Math. 1 (1984), 173-181. MR 839312 (88b:41008)
  • [27] G. Raugel, Résolution numérique par une méthode d'éléments finis du problème de Dirichlet pour le laplacien dans un polygone, C.R. Acad. Sci. Paris Sér. A 286 (1978), 791-794. MR 497667 (81e:65060)
  • [28] P. A. Raviart and J. M. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp. 31 (1977), 391-413. MR 0431752 (55:4747)
  • [29] L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, Modél. Math. Anal. Numér. 19 (1985), 111-143. MR 813691 (87i:65190)
  • [30] G. Szegö, Orthogonal polynomials, Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1978.
  • [31] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 253-272. MR 0463908 (57:3846)
  • [32] M. Vogelius, A right-inverse for the divergence operator in spaces of piecewise polynomials. Application to the p-version of the finite element method, Numer. Math. 41 (1983), 19-37. MR 696548 (85f:65113a)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30

Retrieve articles in all journals with MSC: 65N30


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-0995205-7
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society