In this work three domain decomposition formulations combined with the Preconditioned Conjugate Gradient (PCG) method for solving large-scale linear problems in structural mechanics are presented. In the first approach a subdomain-by-subdomain PCG algorithm is implemented on the global stiffness matrix. The dominant matrix-vector operations are performed localy on the basis of a multi-element group partitioning of the entire domain. The approximate inverse of the global stiffness matrix, which stands as the preconditioner, is expressed by a truncated Neumann series and a lumped preconditioning matrix is formed by its local contributions. In the second approach the PCG algorithm is applied on the interface problem after eliminating the internal degrees of freedom. For this Schur complement implementation the preconditioner is formulated from local Schur complement contributions of each subdomain. The approximate inverse of the global Schur complement which now acts as the preconditioner is also expressed, as in the previous case, by the contributions of the local Schur complements expressed by truncated Neumann series. The third approach, which is a dual domain decomposition formulation, operates on the global level after partitioning the domain into a set of totally disconnected subdomains using Lagrange multipliers. The local problem at each subdomain is solved by the PCG method while the interface problem is handly by a Preconditioned Conjugate Projected Gradient algorithm.

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