A Fourth-Order Cut-cell Multigrid Method for Generic Elliptic Equations on Arbitrary Domains

TL;DR

This work introduces a fourth-order cut-cell geometric multigrid method for solving generic constant-coefficient elliptic equations on arbitrary 2D domains using Cartesian grids. Central to the approach is the Yin-set model of planar continua, which enables robust representation of complex topology, and a cut-cell discretization that combines SFV and PLG cells to achieve high-order accuracy even at ${\mathcal{C}}^1$ discontinuities. The method employs a specialized multigrid solver with an optimal LU factorization of the PLG subblock, a block smoother, and both V- and FMG-cycles to attain the optimal $O(h^{-2})$ complexity, demonstrated across diverse geometries including a panda and rotated squares. Results show the method is accurate, robust, and efficient, with strong conditioning and compatibility with general elliptic forms and boundary conditions, making it suitable for high-fidelity simulations on irregular domains.

Abstract

To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to all forms of elliptic equations and all types of boundary conditions, (2) the versatility of handling both regular and irregular domains with arbitrarily complex topology and geometry, (3) the fourth-order accuracy even at the presence of ${\cal C}^1$ discontinuities on the domain boundary, and (4) the optimal complexity of $O(h^{-2})$. Test results demonstrate the generality, accuracy, efficiency, robustness, and excellent conditioning of the proposed method.

Figures (10)

• Figure 1: An illustration of \ref{['alg:merging']} in generating $\mathsf{C}_{\epsilon}^h(\Omega)$ in (\ref{['eq:mergedCutCellSet']}) with $\epsilon=0.2$ by cutting and merging cells for the domain $\Omega$. The cut cells ${\@fontswitch{}{\mathcal{}} C}_{\mathbf{l}}$, ${\@fontswitch{}{\mathcal{}} C}_{\mathbf{p}}$, and ${\@fontswitch{}{\mathcal{}} C}_{\mathbf{q}}$ are regular, empty, and irregular, respectively. The symbol "$\leftrightarrow$" indicates cell merging. Originally, ${\@fontswitch{}{\mathcal{}} C}_{\mathbf{k}}\in \mathsf{C}_{\Omega}$ is an irregular cell with two small connected components, which are merged to $\mathcal{C}_{\mathbf{j}}$ at line 5 and $\mathcal{C}_{\mathbf{i}}$ at line 9, respectively. Then ${\@fontswitch{}{\mathcal{}} C}_{\mathbf{k}}$ is removed from $\mathsf{C}_{\epsilon}^h(\Omega)$ and its type changed from "irregular" to "empty." The type of ${\@fontswitch{}{\mathcal{}} C}_{\mathbf{i}}$ is changed from "regular" to "irregular."
• Figure 1: An example of ghost filling near the regular boundary. $\mathrm{F}_{\mathbf{i}+\frac{1}{2}\bm{\mathrm{e}}^d}$ is an extendable face and ${\@fontswitch{}{\mathcal{}} C}_{\mathbf{i}}$ is an extendable cell in the high direction along the first dimension.
• Figure 1: Illustrating the total ordering of PLG cells in \ref{['def:linearOrderingOfPLGcells']}. The SFV and PLG cells are shaded in gray and in yellow, respectively. The dashed boxes represent the coarse cells. Fine SFV cells (such as the two adjacent to #14 and #16) may be covered by a coarse PLG cell.
• Figure 1: Results of the proposed cut-cell multigrid method in solving the two equivalent tests in \ref{['sec:squares']} with $h=\frac{1}{256}$. The two exact solutions are related by a rotation of $\frac{\pi}{6}$ around $(0,0)$.
• Figure 2: An example of the stencil for multivariate polynomial fitting in the FV formulation for ${\scriptsize \textsf{D}} = 2$ and $n=4$. The starting point $\mathbf{q}=\mathbf{i}$ is marked by "$\bullet$," the multiindices of shaded cells constitute $\mathcal{S}_{\mathrm{PLG}}$ in (\ref{['eq:PLGstencil']}), and the thick curve segment represents the cut boundary $\mathcal{B}_\mathbf{i}$.

Theorems & Definitions (17)

• Definition 2.1: Yin Space Zhang2020:YinSets
• Theorem 2.2: Zhang and Li Zhang2020:YinSets
• Corollary 2.3
• Definition 4.1
• Definition 4.2: Lagrange interpolation problem (LIP)
• Definition 4.3: PLG in $\mathbb{Z}^{\scriptsize \textsf{D}}$ Zhang:PLG
• Lemma 4.4
• Proof 1
• Definition 5.1
• Lemma 5.2