Spectral Methods for Axisymmetric Domains

Christine Bernardi, Monique Dauge and Yvon Maday

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.

We first present the functional framework for this family of equations: we fully characterize the special weighted spaces on the meridian domain associated with the Fourier coefficients of functions belonging to standard three-dimensional Sobolev spaces. Then starting from a well-posed three-dimensional problem, we write each two-dimensional equation in variational form and prove its well-posedness. When the meridian domain is polygonal, we describe the singular and regular parts of the solutions.

The second step is the discretization of the two-dimensional equations resulting from the Laplace, Stokes and Navier-Stokes equations. We choose spectral methods, since the approximation by high degree polynomials presents the same infinite degree of accuracy as the truncation of Fourier series. We prove error estimates and present some numerical experiments.

In conclusion, this book 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.

Book Series in Applied Mathematics, 3, 1999, 345p.