An explicit norm representation for the analysis of multilevel methods (Q1272510)

The author develops an explicit representation of the norm \[ | | | u| | | ^2 := \inf\Big\{ \sum_i b_i(u_i,u_i): u=\sum_i u_i,\;u_i\in V_i\Big\} \] for a decomposition of some finite dimensional Hilbert space \(V\) into a (not necessarily direct) sum of Hilbert spaces \(V_i\) with inner product \(b_i\). The analysis is motivated by the multilevel method for the numerical solution of symmetric variational problems in \(V\) as described by additive Schwarz schemes, where such a decomposition is used, and where the convergence rate strongly depends on \(| | | \cdot| | | \). The application of this result is demonstrated in detail for the decomposition of sparse grid spaces, and is briefly indicated for approximation spaces and non-nested finite element spaces.