A Geometric Multigrid Preconditioner for Shifted Boundary Method

TL;DR

This paper introduces a geometric multigrid preconditioner with a novel smoother for the Shifted Boundary Method, enabling efficient high-order discretizations by maintaining low iteration counts and mesh independence.

Contribution

It develops a new multigrid preconditioner with a Full-Residual Shy Patch smoother tailored for SBM, improving convergence for high-order discretizations.

Findings

Achieves stable iteration counts up to polynomial degree 3 in 3D

Maintains low mesh dependence and algebraic efficiency

Demonstrates effectiveness for Continuous Galerkin approximations

Abstract

The Shifted Boundary Method (SBM) trades some part of the burden of body-fitted meshing for increased algebraic complexity. While the resulting linear systems retain the standard $\mathcal{O}(h^{-2})$ conditioning of second-order operators, the non-symmetry and non-local boundary coupling render them resistant to standard Algebraic Multigrid (AMG) and simple smoothers for high-order discretisations. We present a geometric multigrid preconditioner that effectively tames these systems. At its core lies the \emph{Full-Residual Shy Patch} smoother: a subspace correction strategy that filters out some patches while capturing the full physics of the shifted boundary. Unlike previous cell-wise approaches that falter at high polynomial degrees, our method delivers convergence with low mesh dependence. We demonstrate performance for Continuous Galerkin approximations, maintaining low and stable iteration counts up to polynomial degree $p=3$ in 3D, proving that SBM can be both geometrically flexible and algebraically efficient.