  This paper gives a geometric description of functional spaces related to
Domain Decomposition techniques for computing solutions of Laplace and
Helmholtz equations. Understanding the geometric structure of these spaces
leads to algorithms for solving the equations. It leads also to a new
interpretation of classical algorithms, enhancing convergence. The algorithms
are given and convergence is proved. Numerical tests are given.
  This is done by building tools enabling geometric interpretations of the
operators related to Domain Decomposition technique. The Despres operators,
expressing conservation of energy for Helmholtz equation, are defined on the
fictitious boundary and their spectral properties proved.It turns to be the key
for proving convergence of the given algorithm for Helmholtz equation in a
non-dissipating cavity.
  Using these tools, one can prove that the Domain Decomposition setting for
the Helmholtz equation leads to an ill-posed problem. Nevertheless, one can
prove that if a solution exists, it is unique. And that the algorithm do
converge to the solution.

Analysis of PDEs;Functional Analysis