A framework for approximation of the Stokes equations in an axisymmetric domain

TL;DR

This paper introduces a framework that reduces the 3D Stokes equations in axisymmetric domains to a series of 2D problems using Fourier expansion, enabling efficient and accurate solutions for regular data.

Contribution

The paper presents a novel Fourier-based reduction method for the Stokes equations in axisymmetric domains, including error analysis and well-posed variational formulations.

Findings

Fourier expansion effectively decouples 3D problems into 2D problems.

Error due to Fourier truncation is quantifiable and manageable.

Small number of 2D problems suffices for regular data.

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.