Higher-Order Generalized Finite Differences for Variable Coefficient Diffusion Operators

TL;DR

This paper introduces a new meshfree discretization method for variable coefficient diffusion operators that maintains high-order accuracy and inherits properties like diagonal dominance from the discrete Laplace operator, applicable to complex interface problems.

Contribution

The authors develop a novel approach combining the discrete Laplace operator with high-order reconstruction functions to discretize variable coefficient diffusion operators in meshfree methods, ensuring accuracy and stability.

Findings

High-order reconstructions preserve the order of accuracy for smooth coefficients.

The new operator inherits diagonal dominance from the Laplace operator.

Demonstrated applicability to interface problems with point clouds.

Abstract

We present a novel approach of discretizing variable coefficient diffusion operators in the context of meshfree generalized finite difference methods. Our ansatz uses properties of derived operators and combines the discrete Laplace operator with reconstruction functions approximating the diffusion coefficient. Provided that the reconstructions are of a sufficiently high order, we prove that the order of accuracy of the discrete Laplace operator transfers to the derived diffusion operator. We show that the new discrete diffusion operator inherits the diagonal dominance property of the discrete Laplace operator. Finally, we present the possibility of discretizing anisotropic diffusion operators with the help of derived operators. Our numerical results for Poisson's equation and the heat equation show that even low-order reconstructions preserve the order of the underlying discrete Laplace operator for sufficiently smooth diffusion coefficients. In experiments, we demonstrate the applicability of the new discrete diffusion operator to interface problems with point clouds not aligning to the interface and numerically show first-order convergence.