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We discuss a generalization of the Cauchy­Riemann equations, which is applicable to biharmonic problems. A first-order system is derived from these equations, and this system is solved using the least squares method with continuous trial functions set up by the finite element method. Optimal rates of convergence in L2(Ω) are proved in regions with smooth boundaries and optimal rates of convergence are proved in H1(Ω) and, using linear triangular elements, observed in L2(Ω) in polygonal regions. [ABSTRACT FROM AUTHOR]

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 Hsieh­Clough­Tocher element, the discrete Kirchhoff triangle, and a class of nonlocking elements for the Reissner­Mindlin 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. [ABSTRACT FROM AUTHOR]