A MULTIGRID REDUCTION FRAMEWORK FOR DOMAINS WITH SYMMETRIES

Divergence constraints are present in the governing equations of numerous physicalphenomena, and they usually lead to a Poisson equation whose solution represents a bottleneck inmany simulation codes. Algebraic multigrid (AMG) is arguably the most powerful preconditionerfor Poisson's equation, and its effectiveness results from the complementary roles played by thesmoother, responsible for damping high-frequency error components, and the coarse-grid correction,which in turn reduces low-frequency modes. This work presents several strategies to make AMG morecompute-intensive by leveraging reflection, translational, and rotational symmetries. AMGR, ourfinal proposal, does not require boundary conditions to be symmetric, therefore applying to a broadrange of academic and industrial configurations. It is based on a multigrid reduction framework thatintroduces an aggressive coarsening to the multigrid hierarchy, reducing the memory footprint, setup,and application costs of the top-level smoother. While preserving AMG's excellent convergence,AMGR allows one to replace the standard sparse matrix-vector product with the more compute-intensive sparse matrix-matrix product, yielding significant accelerations. Numerical experiments onindustrial CFD applications demonstrated up to 70\% speed-ups when solving Poisson's equation withAMGR instead of AMG. Additionally, strong and weak scalability analyses revealed no significantdegradation.