On the Definition of Dirichlet and Neumann Conditions for the Biharmonic Equation and Its Impact on Associated Schwarz Methods

摘要

We showed that using the classical clamped boundary conditions as "Dirichlet" transmission conditions for a Schwarz algorithm applied to the biharmonic equation leads to a convergence that depends on the overlap cubed, see also Brenner (1996) and Shang and He (2009). A better choice of "Dirichlet" conditions involving a Laplacian leads to a convergence that only depends linearly on the overlap, like in the case of Laplace's equation, without additional computational cost, since the Laplacian appearing in this new "Dirichlet" condition is naturally available, for example in a mixed formulation. We then proved that optimized Schwarz methods do not depend on the choice of what the "Dirichlet" condition is, and they all lead to a still substantially better convergence behavior than the classical Schwarz method with the best "Dirichlet" condition. We also found that transmission conditions based on the thin plate model (D_j and N_j for j = 2,4) are inferior in performance compared to the ones coming from the Stokes model (D_j and N_j for j = 1,3).