A Fast Fourth-Order Cut Cell Method for Solving Elliptic Equations in Two-Dimensional Irregular Domains

TL;DR

Solves constant-coefficient elliptic equations in irregular 2D domains with a fast fourth-order cut-cell method. The method combines Yin-space domain representation, poised lattice generation for high-order boundary stencils, weighted least-squares stencil construction, and a geometric multigrid solver to achieve optimal complexity. Numerical tests demonstrate fourth-order accuracy and scalable performance in complex geometries, highlighting the approach's practicality for irregular-domain simulations. The work lays a foundation for extending to variable coefficients and coupled systems such as INSE via the GePUP framework.

Abstract

We propose a fast fourth-order cut cell method for solving constant-coefficient elliptic equations in two-dimensional irregular domains. In our methodology, the key to dealing with irregular domains is the poised lattice generation (PLG) algorithm that generates finite-volume interpolation stencils near the irregular boundary. We are able to derive high-order discretization of the elliptic operators by least squares fitting over the generated stencils. We then design a new geometric multigrid scheme to efficiently solve the resulting linear system. Finally, we demonstrate the accuracy and efficiency of our method through various numerical tests in irregular domains.

Figures (7)

• Figure 1: A Cartesian grid in the cut cell method. Here small cells with volume fraction below 0.3 have been merged. The cut cells $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are interface, pure and empty cells, respectively. The merged cut cells are linked by the symbol "$\leftrightarrow$".
• Figure 1: Illustration of the FMG cycle for a grid with four levels. FMG begins with a descent to the coarsest grid $\Omega^{8h}$; this is represented by the downward dashed line segments. Then the solution is prolongated to $\Omega^{4h}$ and used as the initial guess to the V-cycle on $\Omega^{4h}$. This "prolongation $+$ V-cycle" process is repeated recursively: the prolongation is represented by an upward dashed line and the V-cycles are represented by the solid lines.
• Figure 1: Solving Problem 1 on the grid of $h=1/80$.
• Figure 2: An example of poised lattices covered by dark-shaded cells in finite-volume discretization for ${\scriptsize \textsf{D}} = 2$ and $n=4$. The cell $\mathcal{C}_{\mathbf{i}}$ is marked by a bullet $\bullet$.
• Figure 2: An illustration for the linear order output by Algorithm \ref{['alg:order']}. The regular cells are shaded light gray, while the irregular cells are shaded yellow.

Theorems & Definitions (9)

• Definition 2.1: Yin Space Zhang2020:YinSets
• Definition 2.2
• Theorem 2.3: Zhang and Li Zhang2020:YinSets
• Corollary 2.4
• Definition 2.5: Lagrange interpolation problem (LIP)
• Definition 2.6: PLG in $\mathbb{Z}^{\scriptsize \textsf{D}}$ Zhang:PLG
• Definition 2.7
• Proposition 2.8
• Proof 1