  Matrix-free techniques play an increasingly important role in large-scale
simulations. Schur complement techniques and massively parallel multigrid
solvers for second-order elliptic partial differential equations can
significantly benefit from reduced memory traffic and consumption. The
matrix-free approach often restricts solver components to purely local
operations, for instance, the Jacobi- or Gauss--Seidel-Smoothers in multigrid
methods. An incomplete LU (ILU) decomposition cannot be calculated from local
information and is therefore not amenable to an on-the-fly computation which is
typically needed for matrix-free calculations. It generally requires the
storage and factorization of a sparse matrix which contradicts the low memory
requirements in large scale scenarios. In this work, we propose a matrix-free
ILU realization. More precisely, we introduce a memory-efficient, matrix-free
ILU(0)-Smoother component for low-order conforming finite elements on
tetrahedral hybrid grids. Hybrid grids consist of an unstructured macro-mesh
which is subdivided into a structured micro-mesh. The ILU(0) is used for
degrees-of-freedom assigned to the interior of macro-tetrahedra. This
ILU(0)-Smoother can be used for the efficient matrix-free evaluation of the
Steklov-Poincare operator from domain-decomposition methods. After introducing
and formally defining our smoother, we investigate its performance on refined
macro-tetrahedra. Secondly, the ILU(0)-Smoother on the macro-tetrahedrons is
implemented via surrogate matrix polynomials in conjunction with a fast
on-the-fly evaluation scheme resulting in an efficient matrix-free algorithm.
The polynomial coefficients are obtained by solving a least-squares problem on
a small part of the factorized ILU(0) matrices to stay memory efficient. The
convergence rates of this smoother with respect to the polynomial order are
thoroughly studied.
