A Multivariate Spline based Collocation Method for Numerical Solution of Partial Differential Equations

TL;DR

This paper introduces a multivariate spline collocation method for solving partial differential equations, demonstrating high accuracy and efficiency through extensive 2D and 3D numerical experiments and comparisons with existing methods.

Contribution

It develops a new spline-based collocation approach for PDEs, extending previous methods and showing improved efficiency and comparable or better accuracy.

Findings

Accurately approximates solutions to Poisson and elliptic PDEs.

Performs well in 2D and 3D numerical experiments.

Outperforms existing multivariate spline methods in efficiency and accuracy.

Abstract

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.