Abstract Meshfree methods (MMs) enjoy advantages in discretizing problem domains over mesh-based methods. Extensive progress has been made in the development of the MMs in the last three decades. The commonly used MMs, such as the reproducing kernel particle methods (RKP), the moving least-square methods (MLS), and the meshless Petrov-Galerkin methods, have main difficulties in numerical integration and in imposing essential boundary conditions (EBC). Motivated by conventional finite volume methods, we propose a meshfree finite volume method (MFVM), where the trial functions are constructed through the conventional RKP or MLS procedures, while the test functions are set to be piecewise constants on Voronoi diagrams built on scattered particles. The proposed method possesses three typical merits: (1) the standard Gaussian rules are proven to produce optimal approximation errors; (2) the EBC can be imposed directly on boundary particles; and (3) mass conservation is maintained locally due to its finite volume formulations. Inf-sup conditions for the MFVM are proven in a one-dimensional problem, and are demonstrated numerically using a generalized eigenvalue problem for higher dimensions. Numerical test results are reported to verify the theoretical findings.

中文翻译：

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一种具有最优数值积分和直接施加基本边界条件的无网格有限体积方法

摘要 Meshfree 方法 (MM) 在离散问题域方面比基于网格的方法具有优势。在过去的三年中，MM 的发展取得了广泛的进展。常用的 MM，如再生核粒子方法 (RKP)、移动最小二乘法 (MLS) 和无网格 Petrov-Galerkin 方法，在数值积分和施加基本边界条件 (EBC) 方面存在主要困难。受传统有限体积方法的启发，我们提出了一种无网格有限体积方法（MFVM），其中试验函数通过传统的 RKP 或 MLS 程序构建，而测试函数设置为基于散射粒子的 Voronoi 图上的分段常数。所提出的方法具有三个典型的优点：(1) 证明标准高斯规则会产生最优逼近误差；(2) EBC 可以直接施加在边界粒子上；(3) 由于其有限的体积公式，质量守恒在局部保持。MFVM 的 Inf-sup 条件在一维问题中得到证明，并使用更高维的广义特征值问题进行数值论证。报告了数值测试结果以验证理论结果。