This talk is based on a joint work with Jinsil Lee.
We propose a collocation method based on multivariate polynomial splines over triangulation/tetrahedralization for 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 under the assumption that the solution $u$ is in $H^3(\Omega)$ over the domain $\Omega$ which is of uniformly positive reach. 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 to show that the new method produces a similar and sometimes more accurate approximation in a more efficient fashion.