Smaller stencil preconditioners for linear systems in RBF-FD discretizations

IF 1.7 3区 数学 Q2 MATHEMATICS, APPLIED
Michael Koch, Sabine Le Borne, Willi Leinen
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引用次数: 0

Abstract

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.

Abstract Image

RBF-FD 离散化中线性系统的更小模版预处理器
径向基函数有限差分(RBF-FD)离散化近来已成为偏微分方程(系统)经典有限差分或有限元离散化的替代方法。在本文中,我们将重点讨论如何为由此产生的线性方程组的迭代求解构建先决条件器。在 RBF-FD 中,通过增大模板尺寸可以获得更高的离散化精度。然而,这会导致刚度矩阵的稀疏程度降低,通常也会导致刚度矩阵的条件变差,这对后续的迭代求解器来说都是挑战。我们建议以 RBF-FD 离散化产生的刚度矩阵为基础构建预调节器,与实际待求解系统的刚度矩阵相比,预调节器的模版尺寸更小。在数值结果中,我们重点关注基于多项式增强的多谐花键(PHS)的 RBF-FD 离散化。我们在三维对流扩散方程的求解中说明了较小模版预处理的性能。
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来源期刊
Numerical Algorithms
Numerical Algorithms 数学-应用数学
CiteScore
4.00
自引率
9.50%
发文量
201
审稿时长
9 months
期刊介绍: The journal Numerical Algorithms is devoted to numerical algorithms. It publishes original and review papers on all the aspects of numerical algorithms: new algorithms, theoretical results, implementation, numerical stability, complexity, parallel computing, subroutines, and applications. Papers on computer algebra related to obtaining numerical results will also be considered. It is intended to publish only high quality papers containing material not published elsewhere.
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