  A cornerstone of numerical homogenization is the equivalence of the
microscopic and the macroscopic energy densities, which is referred to as
Hill-Mandel condition. Among these coupling conditions, the cases of periodic,
linear displacement and constant traction conditions are most prominent in
engineering applications. While the stiffness hierarchy of these coupling
conditions is a theoretically established and numerically verified result, very
little is known about the numerical errors and convergence properties for each
of them in various norms. The present work addresses these aspects both on the
macroscale and the microscale for linear as well as quadratic finite element
shape functions. The analysis addresses aspects of (i) regularity and how its
loss affects the convergence behavior on both scales compared with the a priori
estimates, of (ii) error propagation from micro to macro and of (iii) optimal
micro-macro mesh refinement strategy. For constant traction conditions two
different approaches are compared. The performance of a recovery-type error
estimation based on superconvergence is assessed. All results of the present
work are valid for both the Finite Element Heterogeneous Multiscale Method
FE-HMM and for FE$^2$.
