Spectral algorithms for vector elliptic equations in a spherical gap

Abstract

Abstract This paper is devoted to the numerical solution of three-dimensional elliptic equations of a vector field in the region contained between two concentric spheres. The method of truncated series expansion in orthogonal functions is considered using Chebyshev polynomials and vector spherical harmonics. Spectral algorithms are described for solving two-point boundary-value problems of the vector modes of the Poisson, Helmholtz, and biharmonic equations supplemented by boundary conditions typical of fluid dynamical applications. The accuracy of the algorithms is illustrated by computing some analytical single-mode examples. The proposed algorithms when combined with efficient transform techniques can be used to solve multi-mode nonlinear problems.