A Scalable Substructuring Method by Lagrange Multipliers for Plate Bending Problems

• Authors: Jan Mandel;Radek Tezaur;Charbel Farhat
• Venue: SIAM Journal on Numerical Analysis
• Year: 1999

We present a new Lagrange multiplier-based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the finite element tearing and interconnecting substructuring algorithm to the biharmonic equation. The main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations. The resulting method is proved to have a condition number that does not grow with the number of subdomains but rather grows at most polylogarithmically with the number of elements per subdomain. These optimal properties hold for numerous plate bending elements that are used in practice including the Hsieh--Clough--Tocher element, the discrete Kirchhoff triangle, and a class of nonlocking elements for the Reissner--Mindlin plate models. Computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method. Computational efficiency is also demonstrated with the numerical solution in 45 iterations and 105 seconds on a 64-processor IBM SP2 of a plate bending problem with almost one million degrees of freedom.