We consider local multigrid methods for adaptive finite element and adaptive edge element discretized boundary value problems as well as multilevel preconditioned iterative solvers for the finite element discretization of a special class of saddle point problems. The local multigrid methods feature local smoothing processes on adaptively refined meshes and are applied to adaptive P1 conforming finite element discretizations of linear second order elliptic boundary value problems and to adaptive curl-conforming edge element approximations of H(curl)-elliptic problems and the timeharmonic Maxwell equations. On the other hand, the multilevel preconditioned iterative schemes feature block-diagonal or upper block-triangular preconditioned GMRES or BiCGStab applied to the resulting algebraic saddle point problems and preconditioned CG applied to the associated Schur complement system.
As technologically relevant applications of the above methods to problems in electromagnetism and acoustics, we consider the numerical simulation of Logging-While-Drilling tools in oil exploration and the numerical simulation of piezoelectrically actuated surface acoustic waves, respectively.
Keywords: local multigrid methods, adaptively refined meshes, multilevel preconditioners, saddle point problems, Logging-While-Drilling, surface acoustic waves
