RBF-FD discretization of the Oseen equations

• Published in: Journal of computational physics Vol. 542; p. 114375
• Main Authors: Koch, Michael, Le Borne, Sabine
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2025
• Subjects: Meshless methods; Oseen equations; Pressure constraint; RBF-FD; Oseen equations; Pressure constraint; RBF-FD; Meshless methods
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2025.114375

The radial basis function - finite difference (RBF-FD) method is a (meshless) technique for the discretization of differential operators on scattered node sets. In recent years, it has been successfully applied mostly to scalar partial differential equations (PDEs). The extension to the application to the steady state Oseen equations on (several) scattered node sets is not straightforward but requires novel components which are the subject of this paper. We consider the steady-state Oseen equations in three spatial dimensions, and as a radial basis function, we restrict ourselves to the polyharmonic spline (PHS) with polynomial augmentation. However, the following contributions of our paper may also be applied to other model problems and RBFs. In particular, we will consider the selection of two node sets for the two types of unknowns, velocity and pressure, and subsequent (flexible order) RBF-FD discretization of the various differential operators in the coupled system. We discuss variants for the discretization of the pressure constraint as well as the influence of the viscosity parameter on the convergence of the RBF-FD discretization. Finally, we provide numerical tests for the Oseen equations in three dimensions on complex domains using several node arrangements, convection directions and parameters inherent to the PHS RBF-FD method. The tests demonstrate that the proposed method is stable for discretization step widths between hu=0.01 and hu=0.5 and viscosities in the range of 10−3 to 1 not just on the unit cube but also on a more complicated three-dimensional bunny-shaped domain. In particular, for even degrees of polynomial augmentation of the Laplacian (and lower degrees for involved first order differential operators), we can reach convergence of the same (even) order.