New Preprint on the numerical analysis for the Stokes problem

14 April 2026

A recent research paper titled “Numerical analysis for the Stokes problem with non-homogeneous Dirichlet boundary condition” by Thomas Apel, Katharina Lorenz, and Johannes Pfefferer provides new insights into the numerical analysis of  the Stokes problem which is a simple fluid flow model. A key aspect of the approach is the approximation of the Dirichlet boundary data in the  trace space of the finite element space.

One of the central contributions of the paper is the derivation of optimal discretization error estimates. Importantly, the analysis accounts for geometric complexities: in non-convex domains, corner singularities may arise and reduce the achievable approximation order, depending on the maximal interior angle of the domain. The presented theory incorporates these effects and establishes convergence results. In addition, boundary data with low regularity is studied, and a very weak formulation is introduced for cases in which a weak solution does not exist.

Furthermore the importance of the compatibility condition for the boundary data is discussed. While this condition is often considered necessary, the authors show that it is not required for the well-posedness of either weak or very weak formulations. However, it ensures that the solution satisfies the continuity equation in the distributional sense.

Numerical experiments are presented and confirm the theoretical findings.

Thomas Apel, Katharina Lorenz, Johannes Pfefferer: Numerical analysis for the Stokes problem with non-homogeneous Dirichlet boundary condition. arXiv:2604.11356[math.OC]