Smaller stencil preconditioners for linear systems in RBF-FD discretizations

Radial 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.

Subjects

65D12

65D25

65F08

65F10

65F55

65M12

65N22

Iterative solver

Meshfree method

Polyharmonic spline

Polynomial augmentation

Preconditioner

Radial basis function finite difference (RBF-FD)

DDC Class

519: Applied Mathematics, Probabilities

Funding(s)

Projekt DEAL

Publication version

publishedVersion

Lizenz

https://creativecommons.org/licenses/by/4.0/

Name

s11075-024-01835-7.pdf

Type

Main Article

Size

2.71 MB

Format

Adobe PDF

Options

Smaller stencil preconditioners for linear systems in RBF-FD discretizations