Weak and very weak solutions of the Laplace equation and the Stokes system with prescribed regularity

TL;DR

The paper presents a principled method to construct numerical test problems for the Laplace equation and the Stokes system in polygonal domains that realize prescribed regularity via corner singularities with a tunable exponent $λ$. By exploiting fundamental solutions of the form $u(r,\theta)= r^λ Φ(θ)$ (and the Stokes analogues $u(r,\theta)= r^λ U(θ)$, $p(r,\theta)= r^{λ-1} P(θ)$), it derives admissible $λ$ from boundary-condition eigenvalue relations such as $\sin(λω)=±λ\sin ω$, and uses cut-off techniques to realize homogeneous data away from corners. A concrete 2D limit-case example with $ω=\tfrac{3}{2}π$ and $g(r,θ)= r^{-2/3}\sin(2θ/3)$ illustrates how boundary data may be in $L^2$ without producing a very weak solution, highlighting the delicate interaction between boundary regularity and interior singularity. The work also discusses 3D extensions near edges and corners, providing a framework to validate finite element approximations and to study how singularities influence solution spaces and discretization behavior in elliptic problems.

Abstract

To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for elliptic boundary value problems.

Figures (1)

• Figure 1: Illustration of $u^*$