Discrete harmonics for stream function-vorticity Stokes problem

TL;DR

This paper introduces a novel numerical method using discrete harmonic functions to solve the stream function-vorticity Stokes problem, achieving higher accuracy than classical finite element methods.

Contribution

The paper proposes a new approach employing discrete harmonics to improve the convergence order for vorticity in the Stokes problem.

Findings

Achieves order one error in L2 norm of vorticity

Outperforms classical finite element method of degree one

Provides theoretical proof of improved convergence

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.