  We propose an adaptive multigrid preconditioning technology for solving
linear systems arising from Discontinuous Petrov-Galerkin (DPG)
discretizations. Unlike standard multigrid techniques, this preconditioner
involves only trace spaces defined on the mesh skeleton, and it is suitable for
adaptive hp-meshes. The key point of the construction is the integration of the
iterative solver with a fully automatic and reliable mesh refinement process
provided by the DPG technology. The efficacy of the solution technique is
showcased with numerous examples of linear acoustics and electromagnetic
simulations, including simulations in the high-frequency regime, problems which
otherwise would be intractable. Finally, we analyze the one-level
preconditioner (smoother) for uniform meshes and we demonstrate that
theoretical estimates of the condition number of the preconditioned linear
system can be derived based on well established theory for self-adjoint
positive definite operators.
