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A spectral method for the Stokes problem in three-dimensional unbounded domains. (English) Zbl 0971.35006
The author presents a method for solving the Stokes problem in an unbounded domain: the coupling of the transport boundary operator and a spectral method in spherical coordinates. It is an explicit method by the use of vector-valued spherical harmonics. A uniform inf-sup condition is proved, which provides an optimal error estimate.

35A35 Theoretical approximation in context of PDEs
35R30 Inverse problems for PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35C10 Series solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
65T50 Numerical methods for discrete and fast Fourier transforms
Full Text: DOI
[1] Christine Bernardi and Yvon Maday, Approximations spectrales de problèmes aux limites elliptiques, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 10, Springer-Verlag, Paris, 1992 (French, with French summary). · Zbl 0773.47032
[2] C. Canuto, S. I. Hariharan, and L. Lustman, Spectral methods for exterior elliptic problems, Numer. Math. 46 (1985), no. 4, 505 – 520. · Zbl 0548.65082
[3] C. Cohen-Tannoudji, B. Diu and F. Laloë, Mécanique quantique, Hermann, 1977.
[4] Eric Darve, Fast-multipole method: a mathematical study, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 9, 1037 – 1042 (English, with English and French summaries). · Zbl 0889.65116
[5] James R. Driscoll and Dennis M. Healy Jr., Computing Fourier transforms and convolutions on the 2-sphere, Adv. in Appl. Math. 15 (1994), no. 2, 202 – 250. · Zbl 0801.65141
[6] V. Girault, The Stokes problem and vector potential operator in three-dimensional exterior domains: an approach in weighted Sobolev spaces, Differential Integral Equations 7 (1994), no. 2, 535 – 570. · Zbl 0831.35125
[7] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077
[8] Vivette Girault and Adélia Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal. 114 (1991), no. 4, 313 – 333. · Zbl 0731.35078
[9] Laurence Halpern, Spectral methods in polar coordinates for the Stokes problem. Application to computation in unbounded domains, Math. Comp. 65 (1996), no. 214, 507 – 531. · Zbl 0848.65081
[10] B. Hanouzet, Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova 46 (1971), 227 – 272 (French). · Zbl 0247.35041
[11] M. Lenoir and A. Tounsi, The localized finite element method and its application to the two-dimensional sea-keeping problem, SIAM J. Numer. Anal. 25 (1988), no. 4, 729 – 752. · Zbl 0656.76008
[12] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968 (French). · Zbl 0212.43801
[13] B. Mercier and G. Raugel, Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en \?, \? et séries de Fourier en \?, RAIRO Anal. Numér. 16 (1982), no. 4, 405 – 461 (French, with English summary). · Zbl 0531.65054
[14] J.-C. Nedelec, Résolution des Equations de Maxwell par Méthodes Intégrales, Cours de DEA de l’Ecole Polytechnique, 1998.
[15] V. Rokhlin, Solution of acoustic scattering problems by means of second kind integral equations, Wave Motion 5 (1983), no. 3, 257 – 272. · Zbl 0522.73022
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