Document Zbl 1273.65183

Domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations on unbounded domains. (English) Zbl 1273.65183

J. Sci. Comput. 53, No. 2, 451-480 (2012).

Authors’ summary: We develop a domain decomposition spectral method for mixed inhomogeneous boundary value problems of high-order differential equations defined on unbounded domains. We introduce an orthogonal family of new generalized Laguerre functions, with the weight function \(x_\alpha\), \(\alpha\) being any real number. The corresponding quasi-orthogonal approximation and Gauss-Radau type interpolation are investigated, which play important roles in the related spectral and collocation methods. As examples of applications, we propose the domain decomposition spectral methods for two fourth-order problems, and the spectral method with essential imposition of boundary conditions. The spectral accuracy is proved. Numerical results demonstrate the effectiveness of suggested algorithms.

Reviewer: Wilhelm Heinrichs (Essen)

Cited in 10 Documents

MSC:

65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J30	Higher-order elliptic equations

Keywords:

Laguerre quasi-orthogonal approximation; Laguerre-Gauss-Radau type interpolation; domain decomposition spectral methods; high-order problems with mixed inhomogeneous boundary conditions; collocation method; fourth-order problem; numerical results

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[1]	Auteri, F., Parolini, N., Quartapelle, L.: Essential imposition of Neumann condition in Galerkin-Legendre elliptic solver. J. Comput. Phys. 185, 427–444 (2003) · Zbl 1017.65093 · doi:10.1016/S0021-9991(02)00064-5
[2]	Azaiez, M., Shen, J., Xu, C., Zhuang, Q.: A Laguerre-Legendre spectral method for the Stokes problems in a semi-infinite channel. SIAM J. Numer. Anal. 47, 271–292 (2008) · Zbl 1391.76507 · doi:10.1137/070698269
[3]	Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. Techniques of Scientific Computing, vol. 5 pp. 209–486. Amsterdam, Elsevier (1997)
[4]	Bernardi, C., Coppoletta, G., Maday, Y.: Some spectral approximations of multi-dimensional fourth-order problems. Internal report 90021, Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Paris (1990)
[5]	Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2001) · Zbl 0994.65128
[6]	Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods, Fundamentals in Single Domains. Springer, Berlin (2006) · Zbl 1093.76002
[7]	Funaro, D.: Polynomial Approximations of Differential Equations. Springer, Berlin (1992) · Zbl 0774.41010
[8]	Funaro, D., Kavian, O.: Approximation of some diffusion evolution equations in unbounded domains by Hermite function. Math. Comput. 57, 597–619 (1999) · Zbl 0764.35007
[9]	Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM/CBMS, Philadelphia (1977) · Zbl 0412.65058
[10]	Guo, B.-y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998) · Zbl 0906.65110
[11]	Guo, B.-y.: Error estimation of Hermite spectral method for nonlinear partial differential equations. Math. Comput. 68, 1067–1078 (1999) · Zbl 0918.65069 · doi:10.1090/S0025-5718-99-01059-5
[12]	Guo, B.-y., Ma, H.-p.: Composite Legendre-Laguerre approximation in unbounded domains. J. Comput. Math. 19, 101–112 (2001) · Zbl 0980.65135
[13]	Guo, B.-y., Shen, J.: Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Numer. Math. 86, 635–654 (2000) · Zbl 0969.65094 · doi:10.1007/PL00005413
[14]	Guo, B.-y., Wang, T.-j.: Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel. Math. Comput. 78, 129–151 (2009) · Zbl 1198.65200 · doi:10.1090/S0025-5718-08-02152-2
[15]	Guo, B.-y., Xu, C.-l.: Mixed Laguerre-Legendre pseudospectral method for incompressible fluid flow in an infinite strip. Math. Comput. 72, 95–125 (2003) · Zbl 1093.76047
[16]	Guo, B.-y., Zhang, X.-y.: A new generalized Laguerre spectral approximation and its applications. J. Comput. Appl. Math. 181, 342–363 (2005) · Zbl 1072.65155 · doi:10.1016/j.cam.2004.12.008
[17]	Guo, B.-y., Zhang, X.-y.: Spectral method for differential equations of degenerate type by using generalized Laguerre functions. Appl. Numer. Math. 57, 455–471 (2007) · Zbl 1119.65097 · doi:10.1016/j.apnum.2006.07.032
[18]	Guo, B.-y., Wang, L.-l., Wang, Z.-q.: Generalized Laguerre interpolation and pseudospectral method for unbounded domains. SIAM J. Numer. Anal. 43, 2567–2589 (2006) · Zbl 1116.41002 · doi:10.1137/04061324X
[19]	Guo, B.-y., Sun, T., Zhang, C.: Jacobi and Laguerre quasi-orthogonal approximations and related interpolations. Math. Comput. (in press) · Zbl 1264.41008
[20]	Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for CFD, 2nd edn. Oxford Univ. Press, Oxford (2005) · Zbl 1116.76002
[21]	Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal. 38, 1113–1133 (2000) · Zbl 0979.65105 · doi:10.1137/S0036142999362936
[22]	Wang, L.-l., Guo, B.-y.: Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature. J. Approx. Theory 161, 142–173 (2009) · Zbl 1179.47014 · doi:10.1016/j.jat.2008.08.016
[23]	Xu, C.-l., Guo, B.-y.: Laguerre pseudospectral method for nonlinear partial differential equation. J. Comput. Math. 20, 413–428 (2002) · Zbl 1005.65115

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