Convergence in the incompressible limit of the corner singularities

TL;DR

This paper analyzes the convergence of corner singularity expansions in incompressible flow problems, specifically for the penalized Stokes system, providing explicit formulas and convergence results relevant for numerical methods.

Contribution

It develops a detailed corner singularity theory for the Lamé system and applies it to the penalized Stokes problem, including explicit coefficient formulas and convergence analysis.

Findings

Explicit formulas for singular coefficients near re-entrant corners.

Convergence results for singular parts and remainders in penalized Stokes system.

Framework for developing highly accurate numerical schemes.

Abstract

In this paper, we treat the corner singularity expansion and its convergence result regarding the penalized system obtained by eliminating the pressure variable in the Stokes problem of incompressible flow. The penalized problem is a kind of the Lam\'{e} system, so we first discuss the corner singularity theory of the Lam\'{e} system with inhomogeneous Dirichlet boundary condition on a non-convex polygon. Considering the inhomogeneous condition, we show the decomposition of its solution, composed of singular parts and a smoother remainder near a re-entrant corner, and furthermore, we provide the explicit formulae of coefficients in singular parts. In particular, these formulae can be used in the development of highly accurate numerical scheme. In addition, we formulate coefficients in singular parts regarding the Stokes equations with inhomogeneous boundary condition and non-divergence-free property of velocity field, and thus we show the convergence results of coefficients in singular parts and remainder regarding the concerned penalized problem.