Consider a finite element discretization of the Helmholtz equation \(-\nabla^2u - \omega^2 u = f\) in a general domain \(\Omega\). A non-overlapping domain decomposition approach leads to several matrix systems for the discretized solutions on each subdomain, with suitable coupling terms corresponding to the interaction of these contributions at the domain interfaces. The finite element tearing and interconnecting (FETI) approach [see C. Farhat and F.-X. Roux, Comput. Mech. Adv. 2, No. 1, 1–124 (1994; Zbl 0805.73062)] relies on augmented interface operators to improve the overall behavior of such methods.
The authors show that a discrete Steklov-Poincaré operator, implemented as a complete outer Schur complement, is an optimal choice for this purpose. Several algebraic approximations for the Schur complement are discussed. Numerical examples given in the paper demonstrate that a sparse approximation leads to better convergence behavior while a lumped approximation has good scalability with respect to the number of subdomains.