Geometric multigrid method for solving Poisson's equation on octree grids with irregular boundaries

TL;DR

This paper introduces a geometric multigrid solver that effectively handles irregular domain boundaries on octree grids using level set functions, adaptive refinement, and custom stencils, demonstrating robustness and efficiency.

Contribution

It presents a novel multigrid method incorporating irregular boundary conditions on octree grids with adaptive refinement and subgrid boundary localization.

Findings

The method accurately imposes Dirichlet conditions on irregular boundaries.

It demonstrates robust convergence across various test cases.

The approach maintains second-order accuracy away from boundaries.

Abstract

A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree grids with adaptive refinement, a cell-centered discretization and pointwise smoothing. Boundary locations are determined at a subgrid resolution by performing line searches. For grid blocks near the interface, custom operator stencils are stored that take the interface into account. For grid block away from boundaries, a standard second-order accurate discretization is used. The convergence properties, robustness and computational cost of the method are illustrated with several test cases.