This paper presents a meshfree collocation method for solving high order partial differential equations (PDEs). The leading numerical difficulty is the approximation of high order derivatives. To make the approximation simple and efficient, a moving Taylor polynomial (MTP) approximation is presented by using movable expansion point for each sub-domain. Derivatives can be derived straightforward from the corresponding Taylor coefficients, which are determined by solving a weighted least squares problem. A distinct feature of the method is its ability to give the derivatives along with the shape function itself without further cost. To ensure the accuracy of high order approximation, stability of the weighted least squares problems for determining the Taylor coefficients is another issue should be addressed. For this purpose, the basis functions are rescaled by the size of window functions, and QR decomposition is adopted to solve the weighted least squares problems. The collocation method based on this MTP approximation does not require any grid or background cell, so it is a truly meshfree method. When solving the linear algebraic system generated by the MTP collocation method, a preconditioned sparse biconjugate gradients stabilized (BICGSTAB) solver is used to accelerate the computation speed. Numerical tests show that the proposed method is much accurate and efficient for high order PDEs.

本文提出了一种求解高阶偏微分方程（PDE）的无网格搭配方法。主要的数值困难是高阶导数的近似。为了使近似简单有效，通过为每个子域使用可移动扩展点，提出了移动泰勒多项式（MTP）近似。可以直接从相应的泰勒系数中导出导数，这些泰勒系数是通过求解加权最小二乘问题确定的。该方法的一个显着特征是它能够在不增加成本的情况下提供导数以及形状函数本身。为了确保高阶逼近的准确性，应该解决加权最小二乘问题用于确定泰勒系数的稳定性。以此目的，基本函数根据窗口函数的大小进行缩放，并采用QR分解解决加权最小二乘问题。基于这种MTP近似的并置方法不需要任何网格或背景像元，因此它是真正的无网格方法。在求解由MTP搭配方法生成的线性代数系统时，使用预处理的稀疏双共轭梯度稳定（BICGSTAB）求解器可加快计算速度。数值测试表明，该方法对于高阶PDE的精确度和效率均很高。在求解由MTP搭配方法生成的线性代数系统时，使用预处理的稀疏双共轭梯度稳定（BICGSTAB）求解器可加快计算速度。数值试验表明，该方法对高阶PDE的精确度和效率均很高。在求解由MTP搭配方法生成的线性代数系统时，使用预处理的稀疏双共轭梯度稳定（BICGSTAB）求解器可加快计算速度。数值测试表明，该方法对于高阶PDE的精确度和效率均很高。