On the convergence of monolithic multigrid for implicit Runge-Kutta time stepping of finite element problems

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.