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Spectral methods for singular perturbation problems. (English) Zbl 0812.65100

This is a study on spectral discretization methods (where the discretization error tends to zero faster than any power of the grid size) for singular perturbation problems, i.e. homogeneous Dirichlet problems for inhomogeneous second order differential equations with a small parameter at the second-order term in one and two dimensions, e.g. appearing as advection-diffusion equations.
First, stabilization of the spectral method is discussed: The introduction of artificial viscosity stabilizes the scheme but makes the method only first-order accurate. To avoid the loss of the high spectral accuracy, the author proposes a simultaneous change of the (smooth) right-hand side, similarly to the second-order accurate “Mehrstellen”- method due to H. Yserentant [Numer. Math. 34, 171-187 (1980; Zbl 0413.65072)]. Numerical examples substantiate the improvement for mesh sizes with sufficiently high spectral accuracy.
Second, boundary layer effects are treated by domain decomposition. The interface relaxation of the Dirichlet data follows mainly the pattern of D. Funaro, A. Quarteroni and P. Zanolli [SIAM J. Numer. Anal. 25, No. 6, 1213-1236 (1988; Zbl 0678.65082)], especially in the automatic selection of relaxation parameters. The convergence properties are analyzed for one and two dimensions. Finally, an iterative solver is proposed, getting components for a multigrid method in this way.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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References:

[1] C. Canuto: Spectral methods and maximum principle, to appear in Math. Comp. · Zbl 0699.65080
[2] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang: Spectral methods in fluid dynamics. Springer-Verlag, New York-Berlin-Heidelberg, 1988. · Zbl 0658.76001
[3] J. Doerfer: Mehrgitterverfahren bei singulaeren Stoerungen. Master Thesis, Duesseldorf, 1986.
[4] J. Doerfer and K. Witsch: Stable second order discretization of singular perturbation problems using a hybrid technique.
[5] D. Funaro: Computing with spectral matrices. · Zbl 0891.65118
[6] D. Funaro, A. Quarteroni and P. Zanolli: An iterative procedure with interface relaxation for domain decomposition methods. SIAM J. Num. Anal. 25 (1988). · Zbl 0678.65082
[7] W. Hackbusch: Theorie und Numerik elliptischer Differentialgleichungen. Teubner Studienbücher, Stuttgart, 1986. · Zbl 0609.65065
[8] W. Heinrichs: Line relaxation for spectral multigrid methods. J. Comp. Phys. 77 (1988), 166-182. · Zbl 0649.65055
[9] W. Heinrichs: Multigrid methods for combined finite difference and Fourier problems. J. Comp. Phys. 78 (1988), 424-436. · Zbl 0657.65118
[10] T. Meis, U. Markowitz: Numerische Behandlung partieller Differentialgleichungen. Springer-Verlag, Berlin-Heidelberg-New York, 1978. · Zbl 0418.65044
[11] S.A. Orszag: Spectral methods in complex geometries. J. Comp. Phys. 37 (1980), 70-92. · Zbl 0476.65078
[12] H. Yserentant: Die Mehrstellenformeln für den Laplaceoperator. Num. Math. 34 (1980), 171-187. · Zbl 0413.65072
[13] T.A. Zang, Y.S. Wong and M.Y. Hussaini: Spectral multigrid methods for elliptic equations I. J. Comp. Phys. 48 (1982), 485-501. · Zbl 0496.65061
[14] T.A. Zang, Y.S. Wong and M.Y. Hussaini: Spectral multigrid methods for elliptic equations II. J. Comp. Phys. 54 (1984), 489-507. · Zbl 0543.65071
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