  When considered as a standalone iterative solver for elliptic boundary value
problems, the Dirichlet-Neumann (DN) method is known to converge geometrically
for domain decompositions into strips, even for a large number of subdomains.
However, whenever the domain decomposition includes cross-points, i.e.$\!$
points where more than two subdomains meet, the convergence proof does not hold
anymore as the method generates subproblems that might not be well-posed.
Focusing on a simple two-dimensional example involving one cross-point, we
proposed in a previous work a decomposition of the solution into two parts: an
even symmetric part and an odd symmetric part. Based on this decomposition, we
proved that the DN method was geometrically convergent for the even symmetric
part and that it was not well-posed for the odd symmetric part. Here, we
introduce a new variant of the DN method which generates subproblems that
remain well-posed for the odd symmetric part as well. Taking advantage of the
symmetry properties of the domain decomposition considered, we manage to prove
that our new method converges geometrically in the presence of cross-points. We
also extend our results to the three-dimensional case, and present numerical
experiments that illustrate our theoretical findings.
