Local boundary estimates for the Lagrange multiplier discretization of a Dirichlet boundary value problem with application to domain decomposition

We give an estimate on the error resulting from approximating the outer normal derivative of the solution of a second-order partial differential equation with the Lagrange multiplier obtained in using the Lagrange multiplier method for imposing the Dirichlet boundary conditions. We consider both the case of smooth domains and, in view of an application in the framework of domain decomposition, the case of polygonal domains. The estimate given, supported by numerical results, shows that the mesh in the interior of the (sub)domain can be chosen more coarsely than near the boundary, when only a good approximation of the outer normal derivative is needed, as in the case of the evaluation of the Steklov-Poincare operator, or when solving with a Schur complement approach the linear system arising from the three-fields domain decomposition method.