  Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to
simulate complex multiphysics problems with accelerated performance times.
However, mesh-based GNNs require a large number of message-passing (MP) steps
and suffer from over-smoothing for problems involving very fine mesh. In this
work, we develop a multiscale mesh-based GNN framework mimicking a conventional
iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to
mitigate challenges with conventional mesh-based GNNs. We use the framework to
accelerate phase field (PF) fracture problems involving coupled partial
differential equations with a near-singular operator due to near-zero modulus
inside the crack. We define the initial graph representation using all mesh
resolution levels. We perform a series of downsampling steps using Transformer
MP GNNs to reach the coarsest graph followed by upsampling steps to reach the
original graph. We use skip connectors from the generated embedding during
coarsening to prevent over-smoothing. We use Transfer Learning (TL) to
significantly reduce the size of training datasets needed to simulate different
crack configurations and loading conditions. The trained framework showed
accelerated simulation times, while maintaining high accuracy for all cases
compared to physics-based PF fracture model. Finally, this work provides a new
approach to accelerate a variety of mesh-based engineering multiphysics
problems
