Efficient and well-conditioned ghost-point discretization of boundary operators on unfitted domains

TL;DR

This paper introduces a compact, well-conditioned ghost-point discretization method for boundary operators on unfitted domains, enhancing accuracy and efficiency in solving PDEs on irregular geometries.

Contribution

It proposes a boundary operator-based approach using least-squares reconstruction for high-order accurate, compact stencils that improve stability and performance in unfitted boundary methods.

Findings

Maintains high accuracy near complex boundaries.

Improves conditioning of linear systems in ghost-point methods.

Enhances efficiency and stability in large-scale simulations.

Abstract

Unfitted boundary methods are widely used to numerically solve partial differential equations (PDEs) on irregular domains, avoiding the computational burden of generating boundary-conforming grids. In the finite-difference framework, structured Cartesian grids offer advantages such as ease of implementation and efficient parallelization, while geometry is represented implicitly, for instance, through level-set functions. In this setting, ghost point methods are commonly employed to enforce boundary conditions by introducing additional relations between interior and ghost nodes. However, constructing these relations becomes challenging for high-order accurate discretizations, which often rely on wide stencils that can reduce computational efficiency and degrade performance in large-scale parallel simulations. In this work, we investigate alternative ghost-point discretizations based on compact stencils. We introduce a formulation based on a boundary operator that locally approximates the boundary condition near each ghost node, replacing it with linear relations involving both interior and ghost points. The operator is constructed via least-squares reconstruction, allowing flexible stencil configurations while preserving the desired order of accuracy. Several strategies for selecting and adapting compact stencils are proposed, guided by conditioning criteria and iterative refinement procedures to improve global stability. Numerical experiments on various geometries and convection-diffusion regimes demonstrate the effectiveness of the proposed approach, showing that it maintains high accuracy even in the presence of boundary layers and improves stencil compactness and conditioning of the resulting linear systems.