A spectral method for elliptic equations: the Dirichlet problem

Abstract

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.Comment: This is latex with the standard article style, produced using Scientific Workplace in a portable format. The paper is 22 pages in length with 8 figure

A spectral method for elliptic equations: the Neumann problem

“…The construction of our examples is very similar to that given in [1] for the Dirichlet problem. Our first two transformations have been so chosen that we can invert explicitly the mapping , to be able to better construct our test examples.…”

“…The functions S β,m−2 j are spherical harmonic functions and they are orthonormal on the sphere S 2 ⊂ R 3 . See [1,8] for the definition of these functions. In [1] one also finds the quadrature methods which we use to approximate the integrals over B 1 (0) in (14) and (15).…”

“…See [1,8] for the definition of these functions. In [1] one also finds the quadrature methods which we use to approximate the integrals over B 1 (0) in (14) and (15). The functional 2 in (15) is given by ϒ(1, θ, φ))) 1, θ, φ)))dφ dθ (46) where…”

“…Their bibliographies contain references to earlier papers on spectral methods. The present paper is a continuation of the work in [1] in which a spectral method is given for a general elliptic equation with a Dirichlet boundary condition. Our approach is somewhat different than the standard approaches.…”

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier Series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions

“…Atkinson, Hansen and Chien have successfully used Logan-Shepp polynomials to solve the nonlinear Poisson equation (reformulated as an integral equation) and other partial differential equations [7,49,4,5,8]. One strength of their work is that they prove exponentially fast convergence in solving PDEs.…”