Two variational formulations for elastic domain decomposition problems solved by SGBEM enforcing coupling conditions in a weak form

The solution of Boundary Value Problems of linear elasticity using a domain decomposition approach (DDBVPs) is considered. Some theoretical aspects of two new energy functionals, adequate for a formulation of symmetric Galerkin boundary element method (SGBEM) applied to DDBVPs with non-conforming meshes along interfaces, are studied. Considering two subdomains Ω A and Ω B , the first functional, E ( u A , u B ) , is expressed in terms of subdomain displacement fields, and the second one, Π ( u A , u B , t A , t B ) , in terms of unknown displacements and tractions defined on subdomain boundaries. These functionals generalize the energy functionals studied in the framework of the single domain SGBEM, respectively, by Bonnet [Eng Anal Boundary Elem 1995;15:93–102] and Polizzotto [Eng Anal Boundary Elem 1991;8:89–93]. First, it is shown that the solution of a DDBVP represents the saddle point of the functional E. Second, it is shown that the solution of an SGBEM system of boundary integral equations for a DDBVP corresponds to the saddle point of the functional Π . Then, the functional Π is considered for the finite-dimensional spaces of discretized boundary displacements and tractions showing that the solution of the SGBEM linear system of equations represents the saddle point of Π , generalizing in this way the boundary min–max principle, introduced by Polizzotto, to SGBEM solutions of DDBVPs. Finally, a relation between both energy functionals is deduced.