On symmetrizing the ultraspherical spectral method for self-adjoint problems

Abstract

A mechanism is described to symmetrize the ultraspherical spectral method for self-adjoint problems. The resulting discretizations are symmetric and banded. An algorithm is presented for an adaptive spectral decomposition of self-adjoint operators. Several applications are explored to demonstrate the properties of the symmetrizer and the adaptive spectral decomposition.Polynomial approximation theory suggests that we should seek to numerically represent the variable coefficients in Eq. (1) as Chebyshev polynomial expansions [18]. An adaptation of the fast multipole method (FMM) [19] accelerates the Chebyshev-Legendre [20], ultraspherical-ultraspherical [21], and Jacobi-Jacobi [22, §3.3] connection problems to linear complexity in the degree. Provided the Jacobi parameters are not too large, the acceleration of the connection problem enables Jacobi polynomial expansions with nearly the same rapidity as Chebyshev polynomial expansions. Thus in the problems in this paper, the variable coefficients in Eq. (1) are in fact polynomials, but in practice they could very well be good numerical approximations to functions in a particular space. How such approximations, in particular of the leading coefficient p N , affect the spectrum is beyond the scope of this report. A well-known model problemIt is instructive to begin with a well-known model problem for illustrative purposes. 1 In exceptional circumstances, such as when B is zero apart from the first few columns and the nonzero part prepended to L renders the block operator discretization self-adjoint, symmetry may still be within reach. Many common boundary conditions, however, are genuinely infinite-dimensional. This destruction of symmetry has real consequences for the spectra of finite truncations. Whereas Chebyshev and Legendre second-order differentiation matrices by collocation have real spectra [9], it is easy to show by counter-example that finite truncations of ultraspherical discretizations of self-adjoint operators may produce complex spectra.

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