The fictitious-domain method and explicit continuation operators (Q1309057)

The authors first consider a selfadjoint second-order elliptic problem in a complicated two-dimensional domain, embed this domain into a square and construct (for the continuous problem) an explicit iterative procedure. This procedure includes the continuation operator from the original boundary into the original domain. They show convergence and discretize the problem using linear triangular elements. The procedure and its convergence carry over to the discrete case. The peculiarity is here that all essential operations are done on a square grid (divided by the diagonals to become a triangular grid) which is regardless of the original boundary, only the defects are calculated on a boundary-fitted grid. In this iteration (which converges with a convergence factor independent of the stepsizes) a discrete continuation operator is needed which they construct next in an explicit manner. The paper concludes with the construction of a continuation operator in \(W^{1/2} (\Gamma)\) needed for domain decomposition methods as preconditioning operator; here the authors are able to reduce the problem to the case where \(\Gamma\) is a square.