Multigrid preconditioning has proved to deal efficiently with the critical slowing down of standard Krylov solvers. In the literature two slightly different approaches have been developed, referred to as MG-CGR and DD-αAMG. Several libraries implementing multigrid solvers are publicly available for both CPU and GPU codes. In this talk we will consider the QUDA and DDalphaAMG library which implement the MG-GCR and DD-$\alpha$AMG approach, respectively. We will show a comparison between the two multigrid methods and also between CPU and GPU results using twisted mass fermions at the physical point.

Software Development