  In this contribution, we derive a consistent variational formulation for
computational homogenization methods and show that traditional FE2 and IGA2
approaches are special discretization and solution techniques of this most
general framework. This allows us to enhance dramatically the numerical
analysis as well as the solution of the arising algebraic system. In
particular, we expand the dimension of the continuous system, discretize the
higher dimensional problem consistently and apply afterwards a discrete
null-space matrix to remove the additional dimensions. A benchmark problem, for
which we can obtain an analytical solution, demonstrates the superiority of the
chosen approach aiming to reduce the immense computational costs of traditional
FE2 and IGA2 formulations to a fraction of the original requirements. Finally,
we demonstrate a further reduction of the computational costs for the solution
of general non-linear problems.
