Spectral Methods for Axisymmetric Domains (Series in applied mathematics N° 3)

This book is devoted to the mathematical and numerical analysis of partial differential equations set in a three-dimensional axisymmetric domain, that is, a domain generated by rotation of a bidimensional meridian domain around an axis. Thus a three-dimensional axisymmetric boundary value problem can be reduced to a countable family of two-dimensional equations, by expanding the data and unknowns in Fourier series, and an infinite-order approximation is obtained by truncating the Fourier series. Spectral methods for axisymmetric domains contains a deep analysis requiring precise and optimal parameter-dependent estimates, which is aimed at readers interested in mathematical and numerical analysis. In addition, due to the specificity of the geometry, an accurate discretization of a three-dimensional equation is obtained by solving a small number of two-dimensional systems, which is very efficient for many real-life problems and should be of great help for engineers.

Dimension reduction. Functional and Polynomial Tools, and the Laplace Equation. Functions. Orthogonal polynomials. Polynomial approximation in one dimension. Polynomial approximation in two-dimensional domains. Quadrature formulas and polynomial interpolation. Discretization of the Laplace equation in a cylinder. Discretization of the Laplace equation in other geometries. The Stokes and Navier-Stokes Equations. Discretization of the Stokes problem in a cylinder. Discretization of the Stokes problem with domain decomposition. Discretization of the Navier-Stokes equations. Treatment of Complex Geometries. Geometrical extensions. References. Index.