  Multi-level preconditioners for Discontinuous Galerkin (DG) discretizations
are widely used to solve elliptic equations, and a main ingredient of such
solvers is the interpolation operator to transfer information from the coarse
to the fine grid. Classical interpolation operators give continuous
interpolated values, but since DG solutions are naturally discontinuous, one
might wonder if one should not use discontinuous interpolation operators for DG
discretizations. We consider a discontinuous interpolation operator with a
parameter that controls the discontinuity, and determine the optimal choice for
the discontinuity, leading to the fastest solver for a specific 1D symmetric
interior penalty DG discretization model problem. We show in addition that our
optimization delivers a perfectly clustered spectrum with a high geometric
multiplicity, which is very advantageous for a Krylov solver using the method
as its preconditioner. Finally, we show the applicability of the optimal choice
to higher dimensions.
