The paper presents a cheap technique for the approximation of the harmonic extension from the boundary into the interior of a domain with respect to a given differential operator. The new extension operator is based on the hierarchical splitting of the given f.e. space together with smoothing sweeps and an exact discrete harmonic extension on the lowest level and will be used as a component in a domain decomposition (DD) preconditioner. In combination with an additional algorithmical improvement of this DD-preconditioner solution times faster then the previously studied were achieved for the preconditioned parallelized cg-method. The analysis of the new extension operator gives the result that in the 2D-case O(ln(ln(h-1))) smoothing sweeps per level are sufficient to achieve an h-independent behavior of the preconditioned system provided that there exists a spectrally equivalent preconditioner for the modified Schur complement with spectral equivalence constants independent of h.
Keywords: Boundary value problems, Finite element method, Domain decomposition, Preconditioning, Parallel iterative solvers.