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Explore a 15-minute conference talk comparing and combining two powerful multigrid approaches for solving complex numerical problems in Julia. Contrast Algebraic Multigrid (AMG) and Polynomial Multigrid (pMG) methods, understanding how classical AMG excels when properly tuned with appropriate near null spaces but may struggle with higher-order finite element discretizations, while pMG demonstrates superior performance in coarsening high-order polynomial spaces down to first-order systems. Discover how to leverage the strengths of both approaches by implementing AMG as the coarse-level solver within a pMG hierarchy, creating a hybrid method that combines efficient coarsening from pMG with robust coarse solves from AMG. Learn about the practical implementation of this hybrid approach through the FerriteMultigrid.jl package, which builds on Ferrite.jl for finite element infrastructure and utilizes AlgebraicMultigrid.jl for coarse solves. Gain insights into practical multigrid design for vector PDEs and understand how modern Julia package ecosystems can accelerate solver development, making this presentation valuable for those working in numerical linear algebra, preconditioning strategies, or scalable solvers within Julia's scientific computing ecosystem.
