Variational Germano approach for multiscale formulations

In this chapter the recently introduced Variational Germano procedure is revisited. The procedure is explained using commutativity diagrams. A general Germano identity for all types of discretizations is derived. This relation is similar to the Variational Germano identity, but is not restricted to variational numerical methods. Based on the general Germano identity an alternative algorithm, in the context of stabilized methods, is proposed. This partitioned algorithm consists of distinct building blocks. Several options for these building blocks are presented and analyzed and their performance is tested using a stabilized finite element formulation for the convectionU? diffusion equation. Non-homogenous boundary conditions are shown to pose a serious problem for the dissipation method. This is not the case for the leastsquares method although here the issue of basis dependence occurs. The latter can be circumvented by minimizing a dual-norm of the weak relation instead of the Euclidean norm of the discrete residual.