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A Moving Mesh Isogeometric Method Based on Harmonic Maps

Tao Wang, Xucheng Meng, Ran Zhang, Guanghui Hu

TL;DR

The paper addresses the efficiency bottleneck of isogeometric analysis (IGA) by introducing a Moving Mesh Isogeometric Method (MMIGM) that couples high-regularity NURBS-based discretizations with harmonic-map–driven r-adaptive mesh redistribution. By solving a Poisson problem on a moving mesh and using energy-minimizing harmonic maps, MMIGM computes accurate mesh movement via the map gradient $\frac{\partial(x,y)}{\partial(\xi,\eta)}$, enabling flexible monitor functions that incorporate $\nabla u_h$ and higher derivatives. Key contributions include (i) a robust MMIGM framework with explicit mesh-movement formulas, (ii) demonstration that $k$-refinement achieves similar accuracy to $hp$-refinement with far fewer DOFs, (iii) strategies to suppress Gibbs phenomena through mesh redistribution, and (iv) successful extension to three-dimensional all-electron Kohn–Sham simulations for a helium atom. The results indicate improved accuracy and computational efficiency for practical simulations, with potential impact on complex 3D KS and other PDE systems.

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.

A Moving Mesh Isogeometric Method Based on Harmonic Maps

TL;DR

The paper addresses the efficiency bottleneck of isogeometric analysis (IGA) by introducing a Moving Mesh Isogeometric Method (MMIGM) that couples high-regularity NURBS-based discretizations with harmonic-map–driven r-adaptive mesh redistribution. By solving a Poisson problem on a moving mesh and using energy-minimizing harmonic maps, MMIGM computes accurate mesh movement via the map gradient , enabling flexible monitor functions that incorporate and higher derivatives. Key contributions include (i) a robust MMIGM framework with explicit mesh-movement formulas, (ii) demonstration that -refinement achieves similar accuracy to -refinement with far fewer DOFs, (iii) strategies to suppress Gibbs phenomena through mesh redistribution, and (iv) successful extension to three-dimensional all-electron Kohn–Sham simulations for a helium atom. The results indicate improved accuracy and computational efficiency for practical simulations, with potential impact on complex 3D KS and other PDE systems.

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.
Paper Structure (15 sections, 3 theorems, 42 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 3 theorems, 42 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A brief review of isogeometric analysis
  3. The Poisson equation and its isogeometric discretization
  4. A priori error estimate
  5. A moving mesh isogeometric method
  6. Harmonic maps
  7. Framework of MMIGM
  8. Features of MMIGM
  9. Numerical experiments
  10. Case 1: IGA for the Poisson equation
  11. Case 2: gradient in the monitor function
  12. Case 3: high-order derivatives in the monitor function
  13. Case 4: Gibbs phenomenon at the interface
  14. Case 5: the Kohn--Sham equation for a helium atom
  15. Conclusions

Key Result

Theorem 2.1

Let $r$ and $s$ be integers satisfying $0\leq r\leq s\leq p+1$, there exists a positive constant $C_{shape}$ such that where $\mathcal{H}^m$ is the bent Sobolev space of order $m$, $h_K$ denotes the diameter of the element $K$, $\Pi_{\mathcal{N}_h}: L^2(\Omega)\rightarrow \mathcal{N}_h$ is the projector, and $C_{shape}$ is a constant only depending on the geometry of the physical domain $\Omega$.

Figures (9)

  • Figure 1: The fixed logical domain (left) and the exact solution \ref{['ExactSolution1']} (right).
  • Figure 2: The mesh in the physical domain $\Omega$, corresponding numerical solution and error from left to right at initial (top), intermediate (mid) and final (bottom) iteration with gradients in the monitor function.
  • Figure 3: The convergence of the $L^2$ error with respect to the CPU time (s), where the numbers in the figure denote the number of Dofs in each scheme.
  • Figure 4: The mesh in the physical domain $\Omega$, corresponding numerical solution and error from left to right at initial (top) and final (bottom) iteration with second-order derivatives in the monitor function.
  • Figure 5: The numerical solutions obtained by the $k$-refinement, and the $hp$-refinement using IGA with a uniform mesh.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Definition 2.1: Support extension
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3