Using a reduced number of Lagrange multipliers for assembling parallel incomplete field finite element approximations

A domain decomposition algorithm based on a hybrid variational principle was proposed by Farhat and Roux for the parallel finite element solution of self-adjoint elliptic partial differential equations. First, the spatial domain was partitioned into a set of totally disconnected subdomains and an incomplete finite element solution was computed in each of these subdomains. Next, a number of Lagrange multipliers equal to the number of degrees of freedom located at the binding interface were introduced to enforce compatibility constraints between the independent local finite element approximations. For structural and mechanical problems, the resulting algorithm was shown to outperform the conventional method of substructures, especially on parallel processors. Here, the use of a much lower number of Lagrange multipliers for interconnecting the incomplete field finite element solutions is investigated. When accuracy is preserved, this approach drastically reduces the computational complexity of the Schur-complement-like coupling system that is associated with the interface region and significantly enhances the overall performance of the methodology. Finite element procedures for both global and piecewise polynomial approximations of the Lagrange multipliers are derived. Finally, some numerical results obtained for structural example problems that validate the main idea and highlight its advantages are presented.