  Finite element discretization of time dependent problems also require
effective time-stepping schemes. While implicit Runge-Kutta methods provide
favorable accuracy and stability problems, they give rise to large and
complicated systems of equations to solve for each time step. These algebraic
systems couple all Runge-Kutta stages together, giving a much larger system
than for single-stage methods. We consider an approach to these systems based
on monolithic smoothing. If stage-coupled smoothers possess a certain kind of
structure, then the question of convergence of a two-grid or multi-grid
iteration reduces to convergence of a related strategy for a single-stage
system with a complex-valued time step. In addition to providing a general
theoretical approach to the convergence of monolithic multigrid methods,
several numerical examples are given to illustrate the theory show how
higher-order Runge-Kutta methods can be made effective in practice.
