  For non-preconditioned Galerkin systems, the condition number grows with the
number of elements as well as the quotient of the maximal and the minimal
mesh-size. Therefore, reliable and effective numerical computations, in
particular on adaptively refined meshes, require the development of appropriate
preconditioners. We analyze and numerically compare multilevel additive Schwarz
preconditioners for hypersingular integral equations, where 2D and 3D as well
as closed boundaries and open screens are covered. The focus is on a new local
multilevel preconditioner which is optimal in the sense that the condition
number of the corresponding preconditioned system is independent of the number
of elements, the local mesh-size, and the number of refinement levels.
