Axisymmetric Stokes equations in polygonal domains: regularity and finite element approximations

TL;DR

This paper investigates the regularity and finite element approximation of axisymmetric Stokes equations in polygonal domains, addressing singularities and providing optimal mesh strategies with numerical validation.

Contribution

It establishes well-posedness and full regularity in weighted Sobolev spaces for the axisymmetric Stokes problem, and designs graded meshes for optimal finite element approximation.

Findings

Optimal convergence rates achieved with graded meshes

Numerical tests confirm theoretical regularity and approximation results

Weighted Sobolev space regularity addresses domain singularities

Abstract

We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain $\Omega$. In particular, taking into account the singular coefficients in the equation and non-smoothness of the domain, we establish the well-posedness and full regularity of the solution in new weighted Sobolev spaces $\maK^m_{\mu, 1}(\Omega)$. Using our a priori results, we give a specific construction of graded meshes on which the Taylor-Hood mixed method approximates singular solutions at the optimal convergence rate. Numerical tests are presented to confirm the theoretical results in the paper.