直接无网格局部Petrov-Galerkin (DMLPG) 方法：一种广义MLS逼近
Direct meshless local Petrov-Galerkin (DMLPG) method: A generalized MLS approximation
AI总结：
提出一种基于广义移动最小二乘逼近的直接无网格局部Petrov-Galerkin方法，通过直接恢复节点测试泛函避免形函数，降低了计算成本并提高了精度。
摘要：
无网格局部Petrov-Galerkin (MLPG) 方法是一种流行的无网格方法，自90年代末以来已成功用于求解多种边值问题。本文利用广义移动最小二乘 (GMLS) 逼近，提出了一种新的直接MLPG技术，称为DMLPG。遵循无网格方法“完全用节点表示”的原则，广义MLS直接从节点值恢复测试泛函，无需通过形函数。这导致更廉价且更精确的方案。特别是，完全避免形函数使得问题弱形式的数值积分可以在低次多项式上进行，而不是复杂的形函数。因此，标准MLS形函数子程序完全不被调用。数值例子说明了新技术相对于经典MLPG的优越性。在理论方面，本文讨论了替代标准MLPG的新离散格式的稳定性和收敛性。然而，它不涉及MLPG整体的稳定性、收敛性或误差估计，这些应从MLPG文献中获取。
英文摘要：
The Meshless Local Petrov{Galerkin (MLPG) method is one of the popular meshless methods that has been used very successfully to solve several types of boundary value problems since the late nineties. In this paper, using a generalized moving least squares (GMLS) approximation, a new direct MLPG technique, called DMLPG, is presented. Following the principle of meshless methods to express everything "entirely in terms of nodes", the generalized MLS recovers test functionals directly from values at nodes, without any detour via shape functions. This leads to a cheaper and even more accurate scheme. In particular, the complete absence of shape functions allows numerical integrations in the weak forms of the problem to be done over low{degree polynomials instead of complicated shape functions. Hence, the standard MLS shape function subroutines are not called at all. Numerical examples illustrate the superiority of the new technique over the classical MLPG. On the theoretical side, this paper discusses stability and convergence for the new discretizations that replace those of the standard MLPG. However, it does not treat stability, convergence, or error estimation for the MLPG as a whole. This should be taken from the literature on MLPG.
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