A Geometric Multigrid Preconditioner for Discontinuous Galerkin Shifted Boundary Method

Abstract

ABSTRACT I introduce a geometric multigrid preconditioner for the Shifted Boundary Method (SBM) designed to solve PDEs on complex geometries. While the SBM simplifies mesh generation by using a nonconforming background grid, it often results in nonsymmetric linear systems, a complete analysis of which is missing from the literature, especially with regard to the algebraic solution and preconditioning of high‐order versions of SBM. Standard multigrid methods with pointwise smoothers prove ineffective for such systems due to the localized perturbations introduced by the shifted boundary conditions. To address this challenge, we introduce a discontinuous Galerkin (DG) formulation for SBM that enables the design of a cell‐wise multiplicative smoother within an ‐multigrid framework. The element‐local nature of DG methods naturally facilitates cell‐wise correction, which can effectively handle the local complexities arising from the boundary treatment. Numerical results for the Poisson equation demonstrate favorable performance with mesh refinement for linear () and quadratic () elements in both 2D and 3D, with iteration counts showing mild growth. However, challenges emerge for cubic () elements, particularly in 3D, where the current smoother shows reduced effectiveness.

The authors' abstract, as published at the source. International Journal for Numerical Methods in Engineering, 2026 · DOI ↗