Higher-Order Generalized Finite Differences for Variable Coefficient Diffusion Operators

TL;DR

The paper introduces a derived diffusion operator for meshfree generalized finite differences to discretize variable-coefficient diffusion operators without requiring gradients of the diffusivity. By weighting a discrete Laplacian with reconstruction-based diffusivity at midpoints, the method preserves the accuracy of the underlying Laplacian (with an additional error term from the reconstruction) and retains diagonal dominance for stability, enabling robust handling of anisotropic diffusion and diffuse interfaces. Through Poisson and heat equation tests on 2D unstructured point clouds, the derived operator achieves up to fourth-order convergence on smooth diffusivity and demonstrates predictable behavior in interface problems, where higher-order accuracy may require domain decomposition. The work provides a practical, stable framework for accurate diffusion simulations on rough, interface-rich domains, with potential extensions to general anisotropic diffusion and more advanced reconstruction schemes.

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.

Figures (10)

• Figure 1: Analytical solution (left) and diffusivity (right) for test case 1.
• Figure 2: $L^\infty$ error for the solution of test case 1 with MLS based operators depending on the number of points $N$. The dotted lines depict the corresponding reference convergence rate.
• Figure 3: $L^\infty$ error for the solution of test case 1 with derived diffusion operators depending on the number of points $N$. The dotted lines depict the corresponding reference convergence rate.
• Figure 4: Diffusivity $\kappa$ for test case 2.
• Figure 5: $L^\infty$ error for the solution of test case 2 with MLS based operators depending on the number of points $N$. The dotted lines depict the corresponding reference convergence rate.

Theorems & Definitions (11)

• Remark 2.1
• Definition 2.2
• Corollary 2.3
• Corollary 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Corollary 3.5