# h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

## TL;DR

This paper develops h-multigrid preconditioners using agglomeration for discontinuous Galerkin methods to efficiently solve complex incompressible flow problems, demonstrating significant time savings in 2D and 3D cases.

## Contribution

It introduces an agglomeration-based h-multigrid approach with efficient coarse grid operator inheritance for DG discretizations of flow equations.

## Key findings

- Achieves uniform convergence with proper stabilization on coarse levels.
- Demonstrates significant execution time reductions in 2D and 3D simulations.
- Effectively handles complex unstructured meshes and real-life problems.

## Abstract

In this work we exploit agglomeration based $h$-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature $h$-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element $L^2$ projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.