In the paper, we propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for solving Stokes and Navier-Stokes equations. We start with a detailed explanation of the method for the Stokes equation and then extend the study to the Navier-Stokes equations. We shall show that the numerical solution can approximate the exact PDE solution very well over several domains. Then we present several numerical experimental results to demonstrate the performance of the method over the 2D and 3D settings. Also, we apply the IPBM method to our method to find the solution over several curved domains effectively. In addition, we present a comparison with the existing multivariate spline methods in \cite{AL02} and several existing methods 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的多元多项式样条配置方法，用于求解Stokes和Navier-Stokes方程。我们首先详细阐述该方法在Stokes方程中的应用，随后将研究扩展至Navier-Stokes方程。研究表明，数值解能够在多个计算域上很好地逼近精确PDE解。我们进一步给出二维与三维情形下的数值实验，以证明该方法的有效性。此外，通过将IPBM方法应用于本文方法，可有效求解多个弯曲域上的解。最后，与现有文献\cite{AL02}中的多元样条方法及若干其他方法进行对比，结果表明新方法在保持相似精度的同时，有时能以更高效率获得更精确的逼近。