Existing nonlocal diffusion models are mainly classified into two categories: bond-based models, which involve a single-fold integral and usually simulate isotropic diffusion, and state-based models, which contain a double-fold integral and can additionally prototype anisotropic diffusion. While bond-based models exhibit more computational efficiency, they sometimes could be limited in modeling capabilities. In this paper, we successfully develop a novel bond-based nonlocal model for the diffusion process with matrix-valued coefficients in non-divergence form. Our approach incorporates the coefficients into a covariance matrix and employs the multivariate Gaussian function with truncation to define the kernel function, and subsequently model the nonlocal diffusion through the bond-based formulation. We establish the well-posedness of the proposed model along with deriving some of its properties on maximum principle and mass conservation. Furthermore, an efficient linear collocation scheme is designed for numerical solution of our model. Comprehensive experiments in two and three dimensions are conducted to showcase application of the proposed nonlocal model to both isotropic and anisotropic diffusion problems, and to demonstrate high-order accuracy and conditional -convergence of the proposed collocation scheme.

中文翻译：

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一种新型的基于键的非散度矩阵值系数非局部扩散模型及其配置离散化


现有的非局部扩散模型主要分为两类：基于键的模型，涉及单重积分，通常模拟各向同性扩散；基于状态的模型，包含双重积分，并且还可以模拟各向异性扩散。虽然基于键的模型表现出更高的计算效率，但有时它们的建模能力可能受到限制。在本文中，我们成功开发了一种新颖的基于键的非局部模型，用于具有非发散形式的矩阵值系数的扩散过程。我们的方法将系数合并到协方差矩阵中，并采用截断的多元高斯函数来定义核函数，然后通过基于键的公式对非局部扩散进行建模。我们建立了所提出模型的适定性，并推导了其关于最大原理和质量守恒的一些属性。此外，为我们模型的数值求解设计了有效的线性配置方案。进行了二维和三维的综合实验，以展示所提出的非局部模型在各向同性和各向异性扩散问题上的应用，并证明所提出的配置方案的高阶精度和条件收敛性。