Stability of the Stokes projection on weighted spaces and applications
Ricardo G. Duran, Enrique Otarola, Abner J. Salgado
TL;DR
This work proves stability of the finite element Stokes projection in weighted Sobolev spaces ${\mathbf{W}}^{1,p}_0(\omega,\Omega) \times L^p(\omega,\Omega)$ with $\omega \in A_p$, on convex polytopes. The authors develop a discrete weighted inf-sup framework, a quasi-interpolation operator, and a discrete Green’s function analysis to obtain the key bound $\| \nabla {\mathsf{u}}_h \|_{L^p(\omega,\Omega)} + \| {\mathsf{\pi}}_h \|_{L^p(\omega,\Omega)} \lesssim \| \nabla {\mathsf{u}} \|_{L^p(\omega,\Omega)} + \| {\mathsf{\pi}} \|_{L^p(\omega,\Omega)}$, enabling error estimates for Stokes problems with singular forcing. The results are then applied to delta-type sources, establishing $L^p$-error bounds that depend on the weight and the singularity strength, and are demonstrated for standard finite element pairs, such as the mini element and Taylor–Hood pairs. This framework supports accurate FE approximations of Stokes systems with measures as forcing terms, broadening the applicability of weighted FE methods in singular source regimes.
Abstract
We show that, on convex polytopes and two or three dimensions, the finite element Stokes projection is stable on weighted spaces $\mathbf{W}^{1,p}_0(ω,Ω) \times L^p(ω,Ω)$, where the weight belongs to a certain Muckenhoupt class and the integrability index can be different from two. We show how this estimate can be applied to obtain error estimates for approximations of the solution to the Stokes problem with singular sources.