The authors present an implementation of, and studies of the convergence of, multigrid solvers for a generalization of finite elements on sparse grids, applied to the Galerkin discretization of second-order elliptic boundary value problems. Schematic, rather abstract, algorithmic forms are presented for the restriction and prolongation of multigrid operators for these “differential forms.” Attention is paid to the operational complexity of the proposed multigrid solver.

Numerical results, albeit sketchy and stressing convergence rates, are presented for an elliptic variational problem with constant real coefficients in a Q-multigrid cycle, for dimensions equal to 2, 3, and 4. A specific illustrative example, with exact and numerical solutions, and plot of solutions would have enhanced the results presented.