High-order maximum-entropy collocation methods

TL;DR

This paper introduces high-order maximum-entropy collocation methods (HOLMES) for PDE approximation, offering a meshless, efficient, and accurate approach that directly integrates geometric modeling with numerical analysis.

Contribution

The paper develops a novel high-order maximum-entropy collocation framework that is meshless, reduces computational time, and seamlessly integrates geometric modeling with PDE approximation.

Findings

Achieves expected convergence rates in numerical examples

Demonstrates effectiveness on domains with implicit and explicit boundaries

Reduces computational times compared to traditional Galerkin methods

Abstract

This paper considers the approximation of partial differential equations with a point collocation framework based on high-order local maximum-entropy schemes (HOLMES). In this approach, smooth basis functions are computed through an optimization procedure and the strong form of the problem is directly imposed at the collocation points, reducing significantly the computational times with respect to the Galerkin formulation. Furthermore, such a method is truly meshless, since no background integration grids are necessary. The validity of the proposed methodology is verified with supportive numerical examples, where the expected convergence rates are obtained. This includes the approximation of PDEs on domains bounded by implicit and explicit (NURBS) curves, illustrating a direct integration between the geometric modeling and the numerical analysis.