Title: A stencil scaling approach for accelerating matrix-free finite element implementations,
Abstract: We present a novel approach to fast on-the-fly low order finite element
assembly for scalar elliptic partial differential equations of Darcy type with
variable coefficients optimized for matrix-free implementations. Our approach
introduces a new operator that is obtained by appropriately scaling the
reference stiffness matrix from the constant coefficient case. Assuming
sufficient regularity, an a priori analysis shows that solutions obtained by
this approach are unique and have asymptotically optimal order convergence in
the $H^1$- and the $L^2$-norm on hierarchical hybrid grids. For the
pre-asymptotic regime, we present a local modification that guarantees uniform
ellipticity of the operator. Cost considerations show that our novel approach
requires roughly one third of the floating-point operations compared to a
classical finite element assembly scheme employing nodal integration. Our
theoretical considerations are illustrated by numerical tests that confirm the
expectations with respect to accuracy and run-time. A large scale application
with more than a hundred billion ($1.6\cdot10^{11}$) degrees of freedom
executed on 14,310 compute cores demonstrates the efficiency of the new scaling
approach.