A Geometric Multigrid Preconditioner for Shifted Boundary Method

TL;DR

This work presents a geometric multigrid preconditioner that effectively tames linear systems resistant to standard Algebraic Multigrid (AMG) and simple smoothers for high-order discretisations, proving that SBM can be both geometrically flexible and algebraically efficient.

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.

Figures (6)

• Figure 1: Schematic illustrating the background mesh, interior cells (green), the surrogate boundary $\tilde{\Gamma}$ (thick blue line) along the upper boundary of the interior cells, and the true boundary $\partial\Omega$ (red). Figure taken from wichrowski2025geometric.
• Figure 2: Left: Boundaries of a vertex patch. Interior boundaries shown in light gray, and the thick blue segments indicate the exterior faces where SBM boundary conditions are applied. Right: the same vertex patch with a gray frame highlighting which degrees of freedom form the patch subspace; three other patches are indicated with gray dashed outlines, and the central vertices are marked by filled gray squares. The vertices marked with small gray squares may be too shy to form their own patches, depending on the shyness threshold. Note that as long as the shyness threshold is at most 3 all degrees of freedom from those patches will be include in other patches.
• Figure 3: Left: Exemplary numerical solution for the Poisson equation on a unit ball, visualized on the surrogate domain ($\lambda=0.25$), obtained with the shifted boundary method. The true boundary $\Gamma$ is shown in black lines; connections between quadrature points ($p=2$) on the surrogate boundary and their projections on the true boundary are illustrated with grey lines. Note that with $\lambda=0.25$, some parts of the true boundary lie within the surrogate domain. Right: Illustrative distribution of minimum and maximum shift magnitudes (normalized by cell size $h$) on the surrogate boundary for the unit ball problem with $p=3$. The values $\pm\sqrt{2}$ represent the theoretical maximum possible normalized shift magnitude in 2D for a square cell. For $\lambda=0.0$ the maximum shift magnitude was 1.3510, while for $\lambda=0.5$ it was 0.6436 and the minimum -0.6299.
• Figure 4: Iteration counts for multigrid preconditioned GMRES solver using the Full-Residual Shy Patch smoother with varying shyness thresholds equal to 3 on unit ball problem for polynomial degrees $p=1,2,3$. Number of smoothing steps is fixed to 3. Left panel presents results in 2D with shyness $3$, right panel in 3D with shyness $4$. In 3D some data points are missing due to excessive memory requirements.
• Figure 5: Comparison of throughput (DoF/s) for the proposed multigrid preconditioner with Full-Residual Shy Patch smoother with $\xi=3$, $s=3$ (solid lines), against an AMG preconditioner (dashed line) from deal.II's interface to Trilinos. The tests are performed on the unit ball problem for polynomial degrees $p=1,2,3$ with $s=3$ smoothing steps. Left: 2D, Right: 3D. For polynomial degrees higher that 1 AMG did not converge. Timings obtained on a single core of an AMD EPYC 7282.