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A $p$-adaptive treecode solution of the Poisson equation in the general domain

Zixuan Cui, Lei Yang

TL;DR

This work tackles efficient Poisson equation solvers on general domains by introducing a $p$-adaptive treecode. It combines a rigorous error analysis of multipole expansions with a nonuniform order strategy and a Hierarchy Geometry Tree implementation on tetrahedral meshes to adapt the expansion order to local accuracy requirements. The resulting method achieves dramatic reductions in computational cost compared to uniform-order treecodes, while preserving accuracy, and demonstrates competitiveness for demanding problems such as demagnetizing-field calculations in micromagnetics. The framework also includes practical acceleration through direct summation when advantageous and lays groundwork for future adaptive mesh refinement to further boost performance.

Abstract

Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the kernel function is singular. In this paper, a $p$-adaptive treecode algorithm is designed to resolve the efficiency issue for problems defined on a general domain. Such a $p$-adaptive implementation is realized through i). conducting a systematical error analysis for the treecode algorithm, ii). designing a strategy for a non-uniform distribution of the order of multipole expansion towards a given error tolerance, and iii). employing a hierarchy geometry tree structure for coding the algorithm. The proposed $p$-adaptive treecode algorithm is validated by a number of numerical experiments, from which the desired performance is observed successfully, i.e., the computational complexity is reduced dramatically compared with the uniform order case, making our algorithm a competitive one for bottleneck problems such as the demagnetizing field calculation in computational micromagnetics.

A $p$-adaptive treecode solution of the Poisson equation in the general domain

TL;DR

This work tackles efficient Poisson equation solvers on general domains by introducing a -adaptive treecode. It combines a rigorous error analysis of multipole expansions with a nonuniform order strategy and a Hierarchy Geometry Tree implementation on tetrahedral meshes to adapt the expansion order to local accuracy requirements. The resulting method achieves dramatic reductions in computational cost compared to uniform-order treecodes, while preserving accuracy, and demonstrates competitiveness for demanding problems such as demagnetizing-field calculations in micromagnetics. The framework also includes practical acceleration through direct summation when advantageous and lays groundwork for future adaptive mesh refinement to further boost performance.

Abstract

Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the kernel function is singular. In this paper, a -adaptive treecode algorithm is designed to resolve the efficiency issue for problems defined on a general domain. Such a -adaptive implementation is realized through i). conducting a systematical error analysis for the treecode algorithm, ii). designing a strategy for a non-uniform distribution of the order of multipole expansion towards a given error tolerance, and iii). employing a hierarchy geometry tree structure for coding the algorithm. The proposed -adaptive treecode algorithm is validated by a number of numerical experiments, from which the desired performance is observed successfully, i.e., the computational complexity is reduced dramatically compared with the uniform order case, making our algorithm a competitive one for bottleneck problems such as the demagnetizing field calculation in computational micromagnetics.
Paper Structure (14 sections, 2 theorems, 32 equations, 9 figures, 2 algorithms)

This paper contains 14 sections, 2 theorems, 32 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Poisson equation and its treecode solver
  3. Poisson equation
  4. Treecode solver with a uniform order of multipole expansion
  5. Treecode with non-uniform order of multipole expansion
  6. An error analysis for the treecode algorithm
  7. A $p$-adaptive implementation
  8. The flowchart of the algorithm and complexity
  9. A trick on further acceleration with direct summation
  10. Numerical experiments
  11. Treecode 1: $p$-adaptive treecode algorithm
  12. Effect of parameters in $p$-adaptive treecode algorithm
  13. Treecode 2: further acceleration using direct summation
  14. Conclusion

Key Result

Lemma 3.1

Let $\mathbf{x}$ and $\mathbf{y}$ be two distinct points in $\mathbb{R}^3$, and consider the kernel function $1/|\mathbf{x} - \mathbf{y}|$. Suppose $|\mathbf{x}|<|\mathbf{y}|$, the multipole expansion of the kernel function in Cartesian coordinates around $\mathbf{y} = \mathbf{y}_c$ and the Gegenbau where $C_k^{1/2}(\mathbf{y})$ is the Gegenbauer polynomial with $k$th degree.

Figures (9)

  • Figure 1: Treecode structure for a single target element with red background. Gray triangles are neighbor elements and white elements are non-neighbor elements.
  • Figure 2: The well-separated target element and source element. Here, $\mathbf{x}$ and $\mathbf{y}_c$ denote the barycenters of the two elements, $R = \left|\mathbf{x}-\mathbf{y}_c\right|$ is the distance between the barycenters, $\mathbf{y}$ represents the quadrature points in the source element, and $r_y$ is the maximum radius of the source element.
  • Figure 3: The parent tetrahedron (left), the eight child tetrahedrons after once refinement (middle), and additional 16 child tetrahedrons after local refinement (right).
  • Figure 4: The octree data structure for the mesh in the right of Fig. \ref{['mesh']}.
  • Figure 5: Comparison of $L^{2}$ error scaling with $N$ for treecode algorithm using different order distributions, where $N$ denotes the number of elements in $\Omega$.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Definition 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof