Optimal preconditioning for the symmetric and non-symmetric coupling of adaptive finite elements and boundary elements

We analyze a multilevel diagonal additive Schwarz preconditioner for the adaptive coupling of FEM and BEM for a linear 2D Laplace transmission problem. We rigorously prove that the condition number of the preconditioned system stays uniformly bounded, independently of the refinement level and the local mesh-size of the underlying adaptively refined triangulations. Although the focus is on the non-symmetric Johnson-N\'ed\'elec one-equation coupling, the principle ideas also apply to other formulations like the symmetric FEM-BEM coupling. Numerical experiments underline our theoretical findings.