In this article, a novel mesh‐free, moving Kriging (MK) based collocation scheme for the numerical solution of partial differential equations (PDEs) is introduced. In contrast to methods that are based on a Galerkin weak form of the governing PDEs, the MK collocation (MKC) approach, which is strong form based, is truly mesh‐free in the sense that no background mesh is required for numerical integration. In fact, the presented approach does not require the evaluation of any integrals. Since the approximation function in the MK framework can be conditioned on point value‐ as well as derivative‐information, the pointwise exact imposition of essential as well as natural boundary conditions is rendered straightforward. By incorporating an explicit linear basis into the MK framework, the first‐order consistency condition is fulfilled, and thus rigid body motions are captured accurately. Moreover, Kriging functions may be conceived that comply with constraints on higher order derivatives such as the PDE at hand at certain locations. This possibility proves useful in improving the solution accuracy in the vicinity of Dirichlet boundaries. This article provides a study of the method's characteristics by means of 2D linear elasticity examples. It concludes with a suggestion on how to apply MKC to nonlinear PDEs.

中文翻译：

---

混合边界条件下偏微分方程数值解的一种基于强形式的移动Kriging搭配方法

本文介绍了一种新颖的基于无网格的移动Kriging（MK）搭配方案，用于偏微分方程（PDE）的数值解。与基于控制PDE的Galerkin弱形式的方法相比，基于强形式的MK配置（MKC）方法是真正无网格的，因为在数值积分方面不需要背景网格。实际上，所提出的方法不需要评估任何积分。由于MK框架中的逼近函数可以以点值和导数信息为条件，因此基本和自然边界条件的逐点精确加法变得简单明了。通过将明确的线性基础纳入MK框架，可以满足一阶一致性条件，因此，可以准确地捕获刚体的运动。此外，可以构想到符合某些位置上对诸如PDE之类的高阶导数的约束的克里格函数。事实证明，这种可能性有助于提高Dirichlet边界附近的求解精度。本文通过2D线性弹性示例对方法的特性进行了研究。最后提出关于如何将MKC应用于非线性PDE的建议。通过2D线性弹性示例了解其特性。最后提出关于如何将MKC应用于非线性PDE的建议。通过2D线性弹性示例了解其特性。最后提出关于如何将MKC应用于非线性PDE的建议。