A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes
- Apel, Thomas
- Kempf, Volker
- Linke, Alexander
- Merdon, Christian
2010 Mathematics Subject Classification
- 65N30 65N15 65D05
2008 Physics and Astronomy Classification Scheme
- 47.10.ad 47.11.Fg
- Anisotropic finite elements, incompressible Navier--Stokes equations, divergence-free methods, pressure-robustness
Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case.
- IMA J. Numer. Anal., 42 (2022), pp. 392--416 (published online on 14.01.2021), DOI 10.1093/imanum/draa097 .