Discrete fractional Sobolev norms for domain decomposition preconditioning

We present a new approach for preconditioning the interface Schur complement arising in the domain decomposition of second-order scalar elliptic problems. The preconditioners are discrete interpolation norms recently introduced in Arioli & Loghin (2009, Discrete interpolation norms with applications. SIAM J. Numer. Anal., 47, 2924–2951). In particular, we employ discrete representations of norms for the Sobolev space of index 1/2 to approximate the Steklov–Poincaré operators arising from nonoverlapping one-level domain decomposition methods. We use the coercivity and continuity of the Schur complement with respect to the preconditioning norm to derive mesh-independent bounds on the convergence of iterative solvers. We also address the case of nonconstant coefficients by considering the interpolation of weighted spaces and the corresponding discrete norms.