Discrete harmonics for stream function-vorticity Stokes problem

TL;DR

This work addresses the 2D Stokes problem in stream function–vorticity form, where standard linear FEM yields only $O(\sqrt{h})$ convergence for the vorticity. It introduces discrete harmonic functions on the boundary through a harmonic lifting, forming a harmonic-augmented discretization that stabilizes the scheme and yields $O(h)$ convergence for the vorticity in the $L^2$ sense. The authors develop the harmonic lifting framework, interpolation/error estimates, and a stability/convergence theory for both the continuous harmonic method and a fully discrete, recursively refined harmonic approximation with Theorem $\text{th2}$ and Theorem $\text{th3}$. Numerical experiments on a unit square confirm the theoretical improvements, with stabilized runs achieving robust convergence and even suggesting near $O(h^2)$ behavior for the vorticity in practice. The approach enhances the reliability of stream function–vorticity discretizations and points to extensions to more complete Stokes/V-P formulations and 3D domains.

Abstract

We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity. For this problem, the classical finite elements method of degree one converges only to order one-half for the L2 norm of the vorticity. We propose to use harmonic functions to approach the vorticity along the boundary. Discrete harmonics are functions that are used in practice to derive a new numerical method. We prove that we obtain with this numerical scheme an error of order one for the L2 norm of the vorticity.

Figures (12)

• Figure 1: A basis function.
• Figure 2: Division of a triangle of $\mathcal{T}_k$.
• Figure 3: Mesh for discrete harmonics at level $k = 3$.
• Figure 4: Vizualisation of $\eta_{S,3}$ (smiling function)
• Figure 5: Meshes A and B

Theorems & Definitions (38)

• Remark 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Definition 1
• Definition 2
• Remark 2
• Definition 3
• Lemma 1