Non-matching grids and Lagrange multipliers

In this paper we introduce a variant of the three-field formulation where we use only two sets of variables. Considering, to fix the ideas, the homogeneous Dirichlet problem for -Delta u = g in Omega, our variables are i) an approximation psi(h) of u on the skeleton (the union of the interfaces of the sub-domains) on an independent grid (that could often be uniform), and ii) the approximations u(h)(s) of u in each subdomain Omega(s) (each on its own grid). The novelty is in the way to derive, from psi(h), the values of each trace of u(h)(s) on the boundary of each Omega(s). We do it by solving an auxiliary problem on each partial derivativeOmega(s) that resembles the mortar method but is more flexible. Optimal error estimates are proved under suitable assumptions.