# Stabilised Asynchronous Fast Adaptive Composite Multigrid using Additive Damping

## TL;DR

This paper introduces a stabilized, asynchronous additive multigrid method with damping for parallel adaptive meshes, enhancing stability and concurrency over traditional schemes.

## Contribution

It proposes a novel additive multigrid variant with auxiliary damping parameters per level and vertex, improving stability and concurrency on adaptive meshes.

## Key findings

- Achieves high concurrency with additive updates.
- Provides improved stability close to multiplicative schemes.
- Enables efficient parallel adaptive mesh solving.

## Abstract

Multigrid solvers face multiple challenges on parallel computers. Two fundamental ones read as follows: Multiplicative solvers issue coarse grid solves which exhibit low concurrency and many multigrid implementations suffer from an expensive coarse grid identification phase plus adaptive mesh refinement overhead. We propose a new additive multigrid variant for spacetrees, i.e. meshes as they are constructed from octrees and quadtrees: It is an additive scheme, i.e. all multigrid resolution levels are updated concurrently. This ensures a high concurrency level, while the transfer operators between the mesh levels can still be constructed algebraically. The novel flavour of the additive scheme is an augmentation of the solver with an additive, auxiliary damping parameter per grid level per vertex that is in turn constructed through the next coarser level---an idea which utilises smoothed aggregation principles or the motivation behind AFACx: Per level, we solve an additional equation whose purpose is to damp too aggressive solution updates per vertex which would otherwise, in combination with all the other levels, yield an overcorrection and, eventually, oscillations. This additional equation is constructed additively as well, i.e. is once more solved concurrently to all other equations. This yields improved stability, closer to what is seen with multiplicative schemes, while pipelining techniques help us to write down the additive solver with single-touch semantics for dynamically adaptive meshes.