LAUSR

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.