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This work provides a comprehensive exploration of various methods in solving incompressible flows using a projection method, and their relation to the occurrence and management of checkerboard oscillations. It employs an algebraic symmetry-preserving framework, clarifying the derivation and implementation of discrete operators while also addressing the associated numerical errors. The lack of a proper definition for the checkerboard problem is addressed by proposing a physics-based coefficient. This coefficient, rooted in the disparity between the compact- and wide-stencil Laplacian operators, is able to quantify oscillatory solution fields with a physics-based, global, normalised, non-dimensional value. The influence of mesh and time-step refinement on the occurrence of checkerboarding is highlighted. Therefore, single measurements using this coefficient should be considered with caution, as the value presents little use without any context and can either suggest mesh refinement or use of a different solver. In addition, an example is given on how to employ this coefficient, by establishing a negative feedback between the level of checkerboarding and the inclusion of a pressure predictor, to dynamically balance the checkerboarding and numerical dissipation. This method is tested for laminar and turbulent flows, demonstrating its capabilities in obtaining this dynamical balance, without requiring user input. The method is able to achieve low numerical dissipation in absence of oscillations or diminish oscillation on skew meshes, while it shows minimal loss in accuracy for a turbulent test case. Despite its advantages, the method exhibits a slight decrease in the second-order relation between time-step size and pressure error, suggesting that other feedback mechanisms could be of interest.

Divergence constraints are present in the governing equations of numerous physical phenomena, and they usually lead to a Poisson equation whose solution represents a bottleneck in many simulation codes. Algebraic Multigrid (AMG) is arguably the most powerful preconditioner for Poisson's equation, and its effectiveness results from the complementary roles played by the smoother, 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 more compute-intensive by leveraging reflection, translational and rotational symmetries. AMGR, our final proposal, does not require boundary conditions to be symmetric, therefore applying to a broad range of academic and industrial configurations. It is based on a multigrid reduction framework that introduces 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 replacing the standard sparse matrix-vector product with the more compute-intensive sparse matrix-matrix product, yielding significant accelerations. Numerical experiments on industrial CFD applications demonstrated up to 70% speed-ups when solving Poisson's equation with AMGR instead of AMG. Additionally, strong and weak scalability analyses revealed no significant degradation.