Abstract
Let Ω be an open, simply connected, and bounded region in ℝd, d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving the elliptic partial differential equation − Δu + γu = f over Ω with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable, and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.
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References
Atkinson, K., Chien, D., Hansen, O.: A spectral method for elliptic equations: the Dirichlet problem. Adv. Comput. Math. doi:10.1007/s10444-009-9125-8 (to appear)
Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 2nd ed. Springer-Verlag, New York (2005)
Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994)
Bagby, T., Bos, L., Levenberg, N.: Multivariate simultaneous approximation. Constr. Approx. 18, 569–577 (2002)
Canuto, C., Quarteroni, A., Hussaini, My., Zang, T.: Spectral Methods in Fluid Mechanics. Springer-Verlag, New York (1988)
Canuto, C., Quarteroni, A., Hussaini, My., Zang, T.: Spectral Methods—Fundamentals in Single Domains. Springer-Verlag, New York (2006)
Doha, E., Abd-Elhameed, W.: Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials. SIAM J. Sci. Comput. 24, 548–571 (2002)
Dunkl, C., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge Univ. Press, Cambridge MA (2001)
Hansen, O., Atkinson, K., Chien, D.: On the norm of the hyperinterpolation operator on the unit disk and its use for the solution of the nonlinear Poisson equation. IMA J. Numer. Anal. 29, 257–283. doi:10.1093/imanum/drm052 (2009)
Logan, B., Shepp, L.: Optimal reconstruction of a function from its projections. Duke Math. J. 42, 645–659 (1975)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Univ. Press, Cambridge MA (2000)
Mikhlin, S.: The Numerical Performance of Variational Methods. Noordhoff Pub., Groningen (1971)
Ragozin, D.: Constructive polynomial approximation on spheres and projective spaces. Trans. Amer. Math. Soc. 162, 157–170 (1971)
Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)
Shen, J., Wang, L.: Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains. SIAM J. Numer. Anal. 45, 1954–1978 (2007)
Stroud, A.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Inc., Englewood Cliffs, NJ (1971)
Xu, Y.: Lecture notes on orthogonal polynomials of several variables. In: Advances in the Theory of Special Functions and Orthogonal Polynomials, pp. 135–188. Nova, Commack, NY (2004)
Xu, Y.: A family of Sobolev orthogonal polynomials on the unit ball. J. Approx. Theory 138, 232–241 (2006)
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Communicated by Yuesheng Xu.
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Atkinson, K., Hansen, O. & Chien, D. A spectral method for elliptic equations: the Neumann problem. Adv Comput Math 34, 295–317 (2011). https://doi.org/10.1007/s10444-010-9154-3
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DOI: https://doi.org/10.1007/s10444-010-9154-3