  Matrix-free finite element implementations for large applications provide an
attractive alternative to standard sparse matrix data formats due to the
significantly reduced memory consumption. Here, we show that they are also
competitive with respect to the run time in the low order case if combined with
suitable stencil scaling techniques. We focus on variable coefficient
vector-valued partial differential equations as they arise in many physical
applications. The presented method is based on scaling constant reference
stencils originating from a linear finite element discretization instead of
evaluating the bilinear forms on-the-fly. This method assumes the usage of
hierarchical hybrid grids, and it may be applied to vector-valued second-order
elliptic partial differential equations directly or as a part of more
complicated problems. We provide theoretical and experimental performance
estimates showing the advantages of this new approach compared to the
traditional on-the-fly integration and stored matrix approaches. In our
numerical experiments, we consider two specific mathematical models. Namely,
linear elastostatics and incompressible Stokes flow. The final example
considers a non-linear shear-thinning generalized Newtonian fluid. For this
type of non-linearity, we present an efficient approach to compute a
regularized strain rate which is then used to define the node-wise viscosity.
Depending on the compute architecture, we could observe maximum speedups of 64%
and 122% compared to the on-the-fly integration. The largest considered example
involved solving a Stokes problem with 12288 compute cores on the state of the
art supercomputer SuperMUC-NG.
