### 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.