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Item type:Publication, Access status: with files , RBF-FD discretization of the Oseen equationsThe 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.Publicationtype: Journal Article34 22 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Smaller stencil preconditioners for linear systems in RBF-FD discretizationsRadial basis function finite difference (RBF-FD) discretization has recently emerged as an alternative to classical finite difference or finite element discretization of (systems) of partial differential equations. In this paper, we focus on the construction of preconditioners for the iterative solution of the resulting linear systems of equations. In RBF-FD, a higher discretization accuracy may be obtained by increasing the stencil size. This, however, leads to a less sparse and often also worse conditioned stiffness matrix which are both challenges for subsequent iterative solvers. We propose to construct preconditioners based on stiffness matrices resulting from RBF-FD discretization with smaller stencil sizes compared to the one for the actual system to be solved. In our numerical results, we focus on RBF-FD discretizations based on polyharmonic splines (PHS) with polynomial augmentation. We illustrate the performance of smaller stencil preconditioners in the solution of the three-dimensional convection-diffusion equation.Publicationtype: Journal Article25 59