We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to the second-order elliptic PDE in non-divergence form. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the 2D and 3D settings. In addition, we present a comparison with the existing multivariate spline methods in \cite{ALW06} and \cite{LW17} to show that the new method produces a similar and sometimes more accurate approximation in a more efficient fashion.

翻译：本文提出一种基于三角剖分或四面体剖分上多元多项式样条的配置方法，用于偏微分方程的数值求解。我们首先详细阐述该方法在泊松方程中的应用，随后将其推广至非散度形式的二阶椭圆型偏微分方程。结果表明，数值解能够很好地逼近偏微分方程的真解。随后，我们展示了大量二维和三维情形下的数值实验结果，以验证该方法的性能。此外，我们与文献\cite{ALW06}和\cite{LW17}中的现有多元样条方法进行了对比，表明新方法能以更高效率获得与之相近甚至更精确的近似解。