The study investigates a time‑simultaneous multigrid algorithm for convection‑dominated transport problems discretised with Galerkin finite elements and the Crank‑Nicolson time integrator. The authors employ a higher‑order variational multiscale (VMS) stabilization that adds a diffusive term to the variational form, thereby suppressing low‑frequency oscillations that typically arise in Galerkin solutions of advection‑diffusion equations with small diffusion coefficients. The stabilization matrix is constructed from the discrete gradient operator and the mass matrix of the vector‑valued finite element space, resulting in an explicit, diagonal mass matrix that can be inverted efficiently. When the Crank‑Nicolson scheme is applied, the resulting linear system for each time step is assembled into a single block‑tridiagonal matrix that couples all time steps for a given spatial node. This space‑only system has block entries that are lower bidiagonal due to the time integrator, while the overall block structure reflects the spatial finite element discretisation.

To solve this large, coupled system, the authors use a time‑simultaneous multigrid method that is closely related to waveform relaxation. The multigrid hierarchy is built by spatial coarsening only; the time step size is kept fixed on all levels. The global restriction operator is defined as a Kronecker product of the spatial restriction operator and the identity in time, and the prolongation is its transpose. On each level, smoothing is performed with GMRES preconditioned by a block Jacobi operator that consists of the block diagonal of the system matrix. Because each block corresponds to a single spatial node, the preconditioner is highly parallelisable. The coarse‑grid correction is performed on a coarser mesh, and the coarse problem is solved directly. The algorithm is shown to be equivalent to a two‑grid method when the coarse grid is used only once.

Numerical experiments on one‑dimensional advection‑diffusion problems demonstrate that the plain Galerkin discretisation leads to spurious oscillations when the diffusion coefficient is small. The VMS stabilization removes these artifacts and yields a smooth solution. For the pure heat equation the multigrid solver exhibits a bounded number of iterations independent of the mesh size. In contrast, for convection‑dominated cases the iteration count grows unless the VMS term is included. Three choices for the scaling parameter γ in the stabilization term are examined: γ = 2, which keeps the continuous problem unchanged on all levels; γ = 0, a level‑independent option; and γ = 1, which retains a larger stabilization on coarser levels. The γ = 2 choice yields the best convergence behaviour in the tests, reducing the iteration count by roughly a factor of four when the mesh is doubled.

The work was carried out by W. Drews, S. Turek and C. Lohmann at the Institute of Applied Mathematics (LSIII) of TU Dortmund University. The authors developed the discretisation, the stabilization strategy, and the multigrid implementation, and performed the numerical studies. No external funding or specific project timeframe is mentioned in the report.