Abstract
We introduce a stabilized treatment of spectral methods. The condition number of the spectral systems is highly improved. Elliptic and biharmonic problems are considered. Suitable interpolants in the case of inhomogeneous Dirichlet boundary conditions are presented. For a direct solver the improvements with respect to rounding error propagation are numerically demonstrated.
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Brandt, A., Fulton, S. R., and Taylor, G. D. (1985). Improved spectral multigrid methods for periodic elliptic problems,J. Comput. Phys. 58, 96–112.
Canuto, G., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1989).Spectral Methods in Fluid Dynamics, 1st Ed., Springer Series in Computational Physics, Springer-Verlag, Berlin.
Funaro, D., and Heinrichs, W. (1990). Some results about the pseudospectral approximation of one-dimensional fourth-order problems,Numer. Math. 58, 399–418 (1990).
Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia.
Heinrichs, W. (1988a). Line relaxation for spectral multigrid methods.J. Comput. Phys. 77, 166–182.
Heinrichs, W. (1988b). Multigrid methods for combined finite difference and Fourier problems,J. Comput. Phys. 78, 424–436.
Heinrichs, W. (1989). Improved condition number for spectral methods,Math. Comput. 53, 103–119.
Orszag, S. A. (1980). Spectral methods for problems in complex geometries,J. Comput. Phys. 37, 70–92.
Phillips, T. N., Zang, T. A., and Hussaini, M. Y. (1985). Spectral multigrid methods for Dirichlet problems, inMultigrid Methods for Integral and Differential Equations, (D. J. Paddon and H. Holstein, (eds.), Clarendon Press, Oxford, pp. 231–252.
Zang, T. A., Wong, Y. S., and Hussaini, M. Y. (1982). Spectral multigrid methods for elliptic equations I,J. Comput. Phys. 48, 485–501.
Zang, T. A., Wong, Y. S., and Hussaini, M. Y. (1984). Spectral multigrid methods for elliptic equations II,J. Comput. Phys. 54, 489–507.
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Heinrichs, W. Stabilization techniques for spectral methods. J Sci Comput 6, 1–19 (1991). https://doi.org/10.1007/BF01068121
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DOI: https://doi.org/10.1007/BF01068121