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 −∆u = g in Ω, our variables are i) an approximation ψh 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 uh of u in each subdomain Ω (each on its own grid). The novelty is in the way to derive, from ψh, the values of each trace of uh on the boundary of each Ω . We do it by solving an auxiliary problem on each ∂Ω that resembles the mortar method but is more flexible. Optimal error estimates are proved under suitable assumptions.