IS THERE MORE THAN ONE DIRICHLET-NEUMANN ALGORITHM FOR THE BIHARMONIC PROBLEM?

The biharmonic problem is a fourth order partial differential equation and thus requires two boundary conditions, and not just one like in the Laplace case, where the notion of Dirichlet and Neumann boundary conditions comes from. A variational formulation gives an indication of which two conditions one could consider as Neumann, and which two as Dirichlet, and this choice was made in the literature to define Dirichlet-Neumann (and other) domain decomposition algorithms for the biharmonic equation. We show here that if one chooses other sets of two boundary conditions as Dirichlet and Neumann, one can obtain other Dirichlet-Neumann algorithms, and we prove that the classical choice leads to an algorithm with much less favorable convergence characteristics than our new choice. Our proof is based on showing that even optimizing the relaxation matrices (not just scalars) arising in the Dirichlet-Neumann algorithm running on two boundary conditions, the classical choice of Dirichlet and Neumann cannot achieve contraction rates comparable to our new choice, even though mesh independent convergence is achieved. We illustrate our results with numerical experiments, also exploring situations not covered by our analysis, and a simulation of the Golden Gate Bridge.