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A fourth-order, multigrid cut-cell method for solving Poisson's equation in three-dimensional irregular domains

Yixiao Qian, Weizhen Li, Yan Tan, Qinghai Zhang

Abstract

We propose a fourth-order cut-cell method for solving Poisson's equations in three-dimensional irregular domains. Major distinguishing features of our method include (a) applicable to arbitrarily complex geometries, (b) high order discretization, (c) optimal complexity. Feature (a) is achieved by Yin space, which is a mathematical model for three-dimensional continua. Feature (b) is accomplished by poised lattice generation (PLG) algorithm, which finds stencils near the irregular boundary for polynomial fitting. Besides, for feature (c), we design a modified multigrid solver whose complexity is theoretically optimal by applying nested dissection (ND) ordering method.

A fourth-order, multigrid cut-cell method for solving Poisson's equation in three-dimensional irregular domains

Abstract

We propose a fourth-order cut-cell method for solving Poisson's equations in three-dimensional irregular domains. Major distinguishing features of our method include (a) applicable to arbitrarily complex geometries, (b) high order discretization, (c) optimal complexity. Feature (a) is achieved by Yin space, which is a mathematical model for three-dimensional continua. Feature (b) is accomplished by poised lattice generation (PLG) algorithm, which finds stencils near the irregular boundary for polynomial fitting. Besides, for feature (c), we design a modified multigrid solver whose complexity is theoretically optimal by applying nested dissection (ND) ordering method.
Paper Structure (24 sections, 14 theorems, 78 equations, 9 figures, 4 tables, 6 algorithms)

This paper contains 24 sections, 14 theorems, 78 equations, 9 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Roadmap
  3. Yin Space
  4. Grid Construction
  5. Spatial Discretization
  6. Discrete Poisson's Equation
  7. Geometric Characterization
  8. Boundary Fitting
  9. Numerical Cubature
  10. Spatial Discretization
  11. Poised Lattice Generation
  12. Merging Algorithms
  13. Multigrid
  14. Nested Dissection Ordering
  15. A Specific ND Ordering Algorithm
  16. ...and 9 more sections

Key Result

Theorem 2

\newlabelthm:yinSetsFormABooleanAlgebra0 The algebra $\mathbf{Y}:=({\mathbb Y},\ \cup^{\perp\perp},\ \cap,\ \, ^{\perp}, \emptyset,\ \mathbb{R}^3)$ is a Boolean algebra.

Figures (9)

  • Figure 1: The intersection lines of surfaces with the plane $z = z_0+h$.
  • Figure 1: For the finite-difference discretization of a spatial operator at red FD node $\mathbf{x}_j$, we select a poised lattice $\mathcal{T}_{\mathbf{j}} = \{\mathbf{x}_j\}$ in $\Pi_3^3$. The red node and the blue nodes represent $\mathcal{T}_{\mathbf{j}}$ and the ellipsoid represents the irregular boundary.
  • Figure 1: (a) illustrates the partition of a two-dimensional regular domain grid using the finite volume method, where the stencil of cell $\mathbf{i}$ includes its three adjacent cell layers $\{\mathbf{i} \pm \mathbf{e}^d, \mathbf{i} \pm 2\mathbf{e}^{d}, \mathbf{i} \pm 3\mathbf{e}^{d}, d = 1,2\}$. In the initial recursion step $C = P_3$, with $A$ occupying the left part and $B$ the right, respectively. The second recursion assigns $C = P_2$, $A = P_1$. (b) shows the corresponding ordering.
  • Figure 1: Solution and solution errors for sphere with $R = 0.3$, $z = 0.5$, $h = \frac{1}{100}$.
  • Figure 2: Illustration of cell merging.
  • ...and 4 more figures

Theorems & Definitions (28)

  • Definition 1: Yin space zhang2024boolean
  • Theorem 2: Zhang and Li zhang2020boolean
  • Definition 3
  • Theorem 4
  • Theorem 1
  • Proof 1
  • Corollary 2
  • Theorem 3
  • Proof 2
  • Corollary 4
  • ...and 18 more