We analyze a least squares formulation for the numerical solution of second order linear transmission problems in two and three dimensions, which allows jumps on the interface. The second order partial differential equation is rewritten as a first order system in the bounded interior domain, and the unbounded exterior domain is treated by means of boundary integral equations. The least squares functional is given in terms of negative order as well as half integer Sobolev norms, which are computed by using multilevel preconditioners for second order elliptic problems and for Symm's integral equation. As preconditioner we use both multiplicative and additive Schwarz algorithms for multilevel decompositions. The flux variable is discretized by using $L^2$-conforming elements, continuous $H^1$-conforming elements or $H(div)$-conforming Raviart-Thomas elements. The condition numbers of the preconditioned linear systems are bounded or logarithmically growing in case of the h-version. Furthermore we use the Least-Squares functional to derive an error-indicator, by using localization theorems for
negative and fractional Sobolev norms.

Numerical experiments for the h-version as well as for the
adaptive version with various combinations of different elements and
preconditioners confirm our theoretical results.

Contact:Thanh Tran
93857041