NONLOCAL OPTIMIZED SCHWARZ METHOD FOR THE HELMHOLTZ EQUATION WITH PHYSICAL BOUNDARIES

We extend the theoretical framework of nonlocal optimized Schwarz methods as introduced in [X. Claeys, ESAIM Math. Model. Numer. Anal., 55 (2021), pp. 429-448], considering a Helmholtz equation posed in a bounded cavity supplemented with a variety of conditions modeling material boundaries. The problem is reformulated equivalently as an equation posed on the skeleton of a nonoverlapping partition of the computational domain, involving an operator of the form "identity + contraction." The analysis covers the possibility of resonance phenomena where the Helmholtz problem is not uniquely solvable. In case of unique solvability, the skeleton formulation is proved coercive, and an explicit bound for the coercivity constant is provided in terms of the inf-sup constant of the primary Helmholtz boundary value problem.