A Moving Mesh Isogeometric Method Based on Harmonic Maps

TL;DR

This paper introduces a novel moving mesh isogeometric method based on harmonic maps, enhancing efficiency and accuracy in solving PDEs, demonstrated through numerical experiments including quantum simulations.

Contribution

It integrates isogeometric analysis with a harmonic map-based moving mesh technique, improving mesh movement accuracy and flexibility for PDE solutions.

Findings

Enhanced mesh movement accuracy using harmonic maps

Improved efficiency in solving Poisson equations

Successful application to quantum atom simulations

Abstract

Although the isogeometric analysis has shown its great potential in achieving highly accurate numerical solutions of partial differential equations, its efficiency is the main factor making the method more competitive in practical simulations. In this paper, an integration of isogeometric analysis and a moving mesh method is proposed, providing a competitive approach to resolve the efficiency issue. Focusing on the Poisson equation, the implementation of the algorithm and related numerical analysis are presented in detail, including the numerical discretization of the governing equation utilizing isogeometric analysis, and a mesh redistribution technique developed via harmonic maps. It is found that the isogeometric analysis brings attractive features in the realization of moving mesh method, such as it provides an accurate expression for moving direction of mesh nodes, and allows for more choices for constructing monitor functions. Through a series of numerical experiments, the effectiveness of the proposed method is successfully validated and the potential of the method towards the practical application is also well presented with the simulation of a helium atom in Kohn--Sham density functional theory.