A spectral Galerkin approximation of the Orr-Sommerfeld eigenvalue problem in a semi-infinite domain (Q1326476)

The convergence of a Galerkin approximation of the Orr-Sommerfeld eigenvalue problem, which is defined in a semi-infinite domain, is studied theoretically. In case the system of trial functions is based on a composite of Jacobi polynomials and an exponential transform of the semi-infinite domain, the error of the Galerkin approximation is estimated in terms of the transformation parmeter \(a\) and the number \(N\) of trial functions. Finite or infinite-order convergence of the spectral Galerkin method is obtained depending on how the transformation parameter is chosen. If the transformation parameter is fixed, then convergence is of finite order only. However, if \(a\) is varied proportional to \(1/N^ \rho\) with an exponent \(0<\rho<1\), then the approximate eigenvalue converges faster than any finite power of \(1/N\) as \(N \to \infty\). Some numerical examples are given.