Improved stencil selection for meshless finite difference methods in 3D

Article Mathematics, Applied

A meshless finite difference method for elliptic interface problems based on pivoted QR decomposition

Oleg Davydov et al.

Summary: The study proposes a new method to solve elliptic interface problems using a meshless finite difference approach, achieving prescribed consistency order on irregular nodes and placing nodes directly on unfitted interfaces, resulting in sparse system matrices. Experimental results show convergence orders up to O(h(6) for both approximate solution and gradient, as well as robust performance when the interface is known inaccurately.

APPLIED NUMERICAL MATHEMATICS (2021)

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Article Mathematics, Applied

A LEAST SQUARES RADIAL BASIS FUNCTION FINITE DIFFERENCE METHOD WITH IMPROVED STABILITY PROPERTIES

Igor Tominec et al.

Summary: This paper introduces a formulation of the RBF-generated finite difference method in a discrete least squares setting to overcome the nonrobustness issue in the presence of Neumann boundary conditions. The high-order convergence under node refinement is proven, and numerical verification shows that the least squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.

SIAM JOURNAL ON SCIENTIFIC COMPUTING (2021)

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Article Mathematics