  Introducing a coupling framework reminiscent of FETI methods, but here on
abstract form, we establish conditions for stability and minimal requirements
for well-posedness on the continuous level, as well as conditions on local
solvers for the approximation of subproblems. We then discuss stability of the
resulting Lagrange multiplier methods and show stability under a mesh
conditions between the local discretizations and the mortar space. If this
condition is not satisfied we show how a stabilization, acting only on the
multiplier can be used to achieve stability. The design of preconditioners of
the Schur complement system is discussed in the unstabilized case. Finally we
discuss some applications that enter the framework.
