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：我们建议采用基于三角形或四面形的多变量多元样条的同位法方法,用于局部差异方程的数值解决方案。 我们首先详细解释Poisson方程的方法, 然后将研究扩展至第二阶次的椭圆式 PDE, 以非diverence的形式。 我们应显示数字解决方案可以非常准确地接近精确的 PDE 解决方案。 然后我们提出大量数字实验结果, 以显示该方法在 2D 和 3D 设置上的性能。 此外, 我们比较了\ cite ALW06} 和\ cite{LW17} 中现有的多变量样条方法, 以显示该新方法以更有效率的方式产生相似的、 有时更准确的近似值 。
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