Integral method for the Stokes problem

We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity formulation. For this problem, the classical finite elements method of degree one converges only in $\text{O}\text{(}\text{h}\text{)}$ for the quadratic norm of the vorticity, if the domain is convex and the solution regular. We propose to use harmonic functions obtained by a simple layer potential to approach vorticity along the boundary. Numerical results are very satisfying and we prove that this new numerical scheme leads to an error of order $\text{O}\text{(h)}$ for the natural norm of the vorticity and under more regularity assumptions from $\text{O}\text{(h}{}^{\mathrm{3/2}}\text{)}$ to $\text{O}\text{(h}{}^2\text{)}$ for the quadratic norm of the vorticity. To cite this article: T. Abboud et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 71–76