  We present families of space-time finite element methods (STFEMs) for a
coupled hyperbolic-parabolic system of poro- or thermoelasticity.
Well-posedness of the discrete problems is proved. Higher order approximations
inheriting most of the rich structure of solutions to the continuous problem on
computationally feasible grids are naturally embedded. However, the block
structure and solution of the algebraic systems become increasingly complex for
these members of the families. We present and analyze a robust geometric
multigrid (GMG) preconditioner for GMRES iterations. The GMG method uses a
local Vanka-type smoother. Its action is defined in an exact mathematical way.
Due to nonlocal coupling mechanisms of unknowns, the smoother is applied on
patches of elements. This ensures the damping of error frequencies. In a
sequence of numerical experiments, including a challenging three-dimensional
benchmark of practical interest, the efficiency of the solver for STFEMs is
illustrated and confirmed. Its parallel scalability is analyzed. Beyond this
study of classical performance engineering, the solver's energy efficiency is
investigated as an additional and emerging dimension in the design and tuning
of algorithms and their implementation on the hardware.
