Solving Partial Differential Equations on Evolving Surfaces via the Constrained Least-Squares and Grid-Based Particle Method

TL;DR

This paper introduces an advanced numerical framework combining grid-based particle methods and constrained least-squares techniques to accurately solve partial differential equations on evolving surfaces, especially under large deformations.

Contribution

The paper develops a novel framework that enhances the accuracy and stability of PDE solutions on evolving surfaces by integrating improved local reconstruction and a new constrained least-squares formulation.

Findings

Improved accuracy in regions with large curvature during surface evolution.

Enhanced conditioning of discretized matrices for surface differential operators.

Numerical experiments demonstrate the method's effectiveness and robustness.

Abstract

We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We introduce a new component in the local reconstruction step of the algorithm and demonstrate numerically that the modification can improve computational accuracy when a large curvature region is developed during evolution. The method also incorporates a recently developed constrained least-squares ghost sample points (CLS-GSP) formulation, which can lead to a better-conditioned discretized matrix for computing some surface differential operators. The proposed framework can incorporate many methods and link various approaches to the same problem. Several numerical experiments are carried out to show the accuracy and effectiveness of the proposed method.