  In this paper we address three aspects of nonlinear computational
homogenization of elastic solids by two-scale finite element methods. First, we
present a nonlinear formulation of the finite element heterogeneous multiscale
method FE-HMM in a Lagrangean formulation that covers geometrical nonlinearity
and, more general, hyperelasticity. Second, a-priori estimates of FE-HMM, which
exist so far only for the fully linear elastic case in solid mechanics, are
assessed in the regime of nonlinear elasticity. The measured convergence rates
agree fairly well with those of the fully linear regime. Third, we revise the
standard solution algorithm of FE$^2$ which is a staggered scheme in terms of a
nested loop embedding the full solution of the micro problem into one macro
solution iteration step. We demonstrate that suchlike staggered scheme, which
is typically realized by a nested two-level Newton algorithm, can safely and
efficiently be replaced by direct alternations between micro and macro
iterations. The novel algorithmic structure is exemplarily detailed for the
proposed nonlinear FE-HMM, its efficiency is substantiated by a considerable
speedup in numerical tests.
