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的配点法，用于求解Stokes方程和Navier-Stokes方程。我们首先详细阐述该方法在Stokes方程中的实现过程，随后将研究扩展到Navier-Stokes方程。我们将证明，在多个求解域上，数值解能够非常精确地逼近偏微分方程的真实解。接着，我们给出若干数值实验结果表明该方法在二维和三维场景中的性能。此外，我们应用IPBM方法有效求解了多个曲面域上的问题。最后，我们将其与文献\cite{AL02}中现有的多元样条方法及其他若干已有方法进行对比，结果表明新方法能以更高效率获得相似甚至更精确的近似解。