Multigrid p-Robustness at Jacobi Speeds: Efficient Matrix-Free Implementation of Local p-Multigrid Solvers

Abstract: Vertex-patch smoothers are essential for the robust convergence of geometric multigrid methods in high-order finite element applications, yet their adoption is traditionally hindered by the prohibitive cost of solving local patch problems. This paper presents a high-performance, matrix-free implementation of a p-multigrid local solver that dismantles the trade-off between smoothing effectiveness and computational efficiency. We focus on the practical realization of this iterative approach, leveraging sum-factorization and explicit SIMD vectorization to minimize memory footprint and maximize arithmetic throughput. The performance analysis demonstrates that the solver effectively hides data-fetching latencies and maintains optimal $\mathcal{O}(p<sup>d)$ memory scaling, even when dominated by geometric data on distorted meshes. The result is a robust smoother that rivals the execution speed of simple pointwise smoothers while preserving the convergence benefits of patch-based methods.