## Abstract

We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

## 1 Introduction

To determine approximate solutions to fluid flow problems in three-dimensional geometries is a computationally demanding task.
In this paper, we present a framework for efficiently solving the stationary, incompressible Stokes equations in an axisymmetric domain

We use Fourier expansions with respect to the angular variable 𝜃, both of the solution and the data, to reduce the three-dimensional Stokes problem to an equivalent, countable family of decoupled two-dimensional problems (set in Ω) for the Fourier coefficients. A natural way to approximate the three-dimensional problem is then to use Fourier truncation and, to obtain a fully discrete scheme, compute approximate solutions to a finite number of the two-dimensional problems.

This is an established technique to approximate boundary value problems that are invariant by rotation. Early error analysis results for second-order elliptic problems can be found in [8] and for Poisson’s equation in domains with reentrant edges in [6], both using finite element approximation for the two-dimensional problems. We refer to [5] for additional references to problems described by Laplace or wave equations, the Lamé system, Stokes or Navier–Stokes systems, and Maxwell’s equations, and to [6] for early references to algorithms and applications.

We analyze the error due to Fourier truncation and show that, for data that are sufficiently regular with respect to 𝜃, it suffices to solve a small number of two-dimensional problems, which makes the method efficient. Also, the decoupling of the two-dimensional problems makes it suitable for parallel implementation. A further advantage is simplification of the mesh-generation, which is only required for the two-dimensional half section Ω.

An added complexity is that the natural, variational spaces for the Fourier coefficients turn out to be weighted Sobolev spaces, where the weight is either the distance to the symmetry axis, or its inverse.
We derive these spaces by decomposing (through a change of variables to cylindrical coordinates) the three-dimensional norms for the relevant spaces

The characterizations of the three-dimensional spaces thus obtained are in agreement with the results in [3], where characterizations of

As recently shown in [5], the more direct approach for Sobolev spaces of integer order (based on changing to cylindrical coordinates in the three-dimensional norms) results in equivalent norms (compared with the norms in [3]), but where the equivalence constants (unlike in [3]) are independent of the domain.
In [5], characterizations of

The purpose of the present work is to give a comprehensive presentation directly aimed at the Stokes problem providing, inter alia, detailed derivations of the relevant two-dimensional spaces and norms. Taking as starting-point, in fact, Fourier decompositions of the three-dimensional inner products

additionally enables us to derive a decomposition of the negative norm

Examples of how to build on this framework by discretizing the two-dimensional problems can be found in [3], where spectral methods are used, and [2], where two families of finite elements of order 2 (one with continuous pressure corresponding to the Taylor–Hood element and one with discontinuous pressure) are used. The case with an axisymmetric solution (where only the Fourier coefficient of order 0 is considered, and the angular velocity component is equal to zero), has been treated with finite elements in [1, 9, 10].

A paper in preparation will be devoted to design and analysis of stabilized finite elements for the two-dimensional problems.

An outline of the paper is as follows.

In Section 2, we give some examples of axisymmetric domains and state the stationary, incompressible Stokes equations.

In Section 3, we recall some basic formulas and state the Stokes problem in cylindrical coordinates.

In Section 4, we use Fourier expansion with respect to the angular variable to reduce the three-dimensional Stokes problem to a countable family of two-dimensional problems.

In Section 5, we derive natural variational spaces for the Fourier coefficients by decomposing the relevant three-dimensional norms into sums over all wavenumbers.

In Section 6, we state variational formulations of the two-dimensional problems and show that these are well-posed.

In Section 7, we introduce two families of anisotropic spaces that we need to analyze the error due to Fourier truncation.

In Section 8, we prove an error estimate due to Fourier truncation.

## 2 Model Description

We consider fluid flow in a bounded domain

### 2.1 Axisymmetric Domains

An example of an axisymmetric domain

In [3],

### 2.2 Stokes Problem

We model fluid flow through an axisymmetric domain

where the unknowns are the velocity

and the data are the source term

where

We recall the standard definitions of the Lebesgue and Sobolev spaces (with all derivatives being taken in the sense of distributions)

and the corresponding inner products

We consider spaces of complex-valued functions.
In Section 4, we will use Fourier expansions of the data and the unknowns to reduce the three-dimensional Stokes problem in

We also recall the subspaces

and the dual of

By the dual space

On

which is a norm, equivalent to

Denoting by

with norm

## 3 Cylindrical Coordinates

Since the domain

We define

and, by analogy,

where

### 3.1 Basic Formulas

We recall the identities (see Figure 4 for an illustration in the

relating the component vectors

expressed in the two coordinate systems.
We will write (3.3) in matrix form

From these identities follow the formulas for the gradient and the Laplacian operator acting on a scalar function

### 3.2 Stokes Problem in Cylindrical Coordinates

Expressing both the data and the unknowns in cylindrical coordinates

where, from (3.5),

## 4 Fourier Expansion

A natural way to reduce the three-dimensional Stokes problem (3.8) in

### 4.1 Two-Dimensional Problems

Inserting the Fourier expansions (4.1)–(4.4) into (3.8) results, since the Stokes problem is linear and invariant by rotation (which means that the coefficients of the Stokes operator in cylindrical coordinates do not depend on 𝜃), in uncoupled two-dimensional problems for each Fourier coefficient pair

where

and

We will, for all

By taking the complex conjugate of (4.5), it is easy to see that, for real-valued data

the pair

(corresponding, of course, to a unique, real-valued solution of the three-dimensional Stokes problem for real-valued data), so in the practical case with real-valued data, we only need to solve the problems (4.5) for

The compatibility condition (2.2) translates into a condition on the Fourier coefficient

where

## 5 Variational Spaces

In Section 6.1, we will state variational formulations of the two-dimensional problems (4.5).
To determine natural variational spaces for the Fourier coefficients defined on the half section Ω, we start by expressing the

Based on the structure of the different terms, which are weighted integrals over Ω of the Fourier coefficients and their derivatives, we define weighted Sobolev spaces on Ω.

As a result, we obtain characterizations of the three-dimensional spaces

Using the results for

### 5.1 Fourier Decomposition of Inner Products and Norms

In cylindrical coordinates, the

is expressed as an integral over

For two vector functions

the

can, through repeated use of relations (3.2), (3.3) and the Pythagorean trigonometric identity, be expressed in cylindrical coordinates:

We now consider Fourier expansions

and, similarly, in four other representative cases, (5.8)

The corresponding decompositions of the associated norms are

### 5.2 Weighted Sobolev Spaces on Ω

Led by (5.10) and (5.11), where each term is an integral over Ω with weight 𝑟 or

Next, we define

and norm

The definition can be extended in a natural way to

We will also need the weighted space

equipped with the norm

It can be proved [8, Proposition 4.1] that all functions in

We finally introduce the subspaces

consisting of functions in

consisting of functions in

All spaces defined above are Hilbert spaces for the inner products associated with the given norms.

### 5.3 Characterization of
L
2
(
Ω
˘
)
and
L
0
2
(
Ω
˘
)

Recalling identity (5.10), we let the right-hand side terms, for all

Introducing the

and combining (5.10) with (5.13)–(5.14), we obtain the following characterization of

The mapping

where, for all

Since, for

we also, recalling (5.12), obtain a characterization of the subspace

The mapping

and, for all

### 5.4 Characterization of
(
H
1
(
Ω
˘
)
)
3
and
(
H
0
1
(
Ω
˘
)
)
3

Similarly, led by the right-hand side terms in identity (5.11), for all

To see that (5.16) satisfies the properties of a norm, we consider the function

and, from (5.11), note that

From (5.17) and the norm properties of

We now characterize the spaces