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 ΩAΩA and ΩBΩB, the first functional, E(uA,uB)E(uA,uB), is expressed in terms of subdomain displacement fields, and the second one, Π(uA,uB,tA,tB)Π(uA,uB,tA,tB), 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 ΠΠ. Then, the functional ΠΠ 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 ΠΠ, 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.