Abstract
A variational spectral element multigrid algorithm is proposed, and results are presented for a one-dimensional Poisson equation on a finite interval. The key features of the proposed algorithm are as follows: the nested spaces and associated hierarchical bases are intra-element, resulting in simple data structures and rapid tensor-product sum-factorization evaluations; smoothing is effected by readily constructed and efficiently inverted (diagonal) Jacobi preconditioners; the technique is readily parallelized within the context of a medium-grained paradigm; and the (work-deflated) multigrid convergence rate\(\bar \rho \) is bounded from above well below unity, and is only a weak function of the number of spectral elementsK, the (large) order of the polynomial approximation,N, and the number of multigrid levels,J. Preliminary tests indicate that these convergence properties persist in higher space dimensions.
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References
Babuska, I., and Dorr, M. R. (1981). Error estimates for the combinedh- and p-version of the finite element method.Numer. Math. 37, 257–277.
Bank, R. E., and Douglas, C. C. (1985). Sharp estimates for multigrid rates of convergence with general smoothing and acceleration.SIAM J. Numer. Anal. 22, 617–633.
Brand, K., Lemke, M., and Linden, J. (1986). Multigrid Bibliography. Arbeitspapiere der GMD (206). Gesellschaft für Mathematik und Datenverarbeitung.
Brandt, A. (1977). Multi-level adaptive solutions to boundary value problems.Math. Comp. 31, 330–390.
Deville, M. O., and Mund, E. H. (1985). Chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning.J. Comput. Phys. 60, 517–533.
Fischer, P., Rønquist, E. M., Dewey, D., and Patera, A. T. (1988). Spectral element methods: Algorithms and architectures. InProceedings of the First International Conference on Domain Decomposition Methods for Partial Differential Equations, Paris, Glowinski, R., Golub, G., Meurant, G., and Periaux, J. (eds.), 173–197. SIAM, Philadelphia.
Gottlieb, D., and Lustman, L. (1983). The spectrum of the Chebyshev collocation operator for the heat equation,SIAM J. Numer. Anal. 20, 909–921.
Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods. SIAM, Philadelphia.
Hackbusch, W., and Trottenberg, U. (eds.) (1982).Multigrid Methods. Proceedings of the Conference Held at Koln-Porz, November 23–27, 1981. Springer-Verlag, New York.
Maday, T., and Munoz, R. (1988). Spectral element multigrid. Part II: Theoretical justification.J. Sci. Comp. (submitted).
Maday, Y., and Patera, A. T. (1987). Spectral element methods for the incompressible Navier-Stokes equations. InState of the Art Surveys in Computational Mechanics, Noor, A. K. (ed.) (to appear). ASME, New York.
Maitre, J. F., and Musy, F. (1983). Multigrid Methods: Convergence Theory in a Variational Framework. Technical Report 20, Université de Lyon.
McBryan, O. A., and Van de Velde, E. F. (1985). The multigrid method on parallel processors. InMultigrid Methods II. Proceedings of the 2nd European Conference on Multigrid Methods, Dold, A., and Eckmann, B. (eds.), 232–260. Springer-Verlag, New York.
Orszag, S. A. (1980). Spectral methods for problems in complex geometries.J. Comput. Phys. 37, 70–92.
Patera, A. T. (1984). A spectral element method for fluid dynamics; Laminar flow in a channel expansion.J. Comput. Phys. 54, 468–488.
Phillips, T. N. (1987). Relaxation schemes for spectral multigrid methods.J. Comput. Appl. Math. 18, 149–162.
Rønquist, E. M., and Patera, A. T. (1987a). A Legendre spectral element method for the Stefan problem.Int. J. Num. Methods Eng. 24, 2273–2299.
Rønquist, E. M., and Patera, A. T. (1987b). A Legendre spectral element method for the incompressible Navier-Stokes equations. InProceedings of the Seventh GAMM Conference on Numerical Methods in Fluid Mechanics (to appear). Vieweg.
Stroud, A. H., and Secrest, D. (1966).Gaussian Quadrature Formulas. Prentice Hall, Englewood Cliffs, New Jersey.
Weideman, J. A. C., and Trefethen, L. N. (1987). The eigenvalues of second-order spectral differentiation matrices.SIAM J. Numer. Anal. (submitted).
Zang, T. A., Wong, Y. S., and Hussaini, M. Y. (1982a). Spectral multigrid methods for elliptic equations.J. Comput. Phys. 48, 485–501.
Zang, T. A., Wong, Y. S., and Hussaini, M. Y. (1982b). Spectral multigrid methods for elliptic equations II.J. Comput. Phys. 54, 489–507.
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Rønquist, E.M., Patera, A.T. Spectral element multigrid. I. Formulation and numerical results. J Sci Comput 2, 389–406 (1987). https://doi.org/10.1007/BF01061297
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DOI: https://doi.org/10.1007/BF01061297