A Least-squares Method for Consistent Mesh Tying

In the finite element method, a standard approach to mesh tying is to apply Lagrange multipliers. If the interface is curved, however, discretization generally leads to adjoining surfaces that do not coincide spatially. Straightforward Lagrange multiplier methods lead to discrete formulations failing a first-order patch test [12]. A least-squares method is presented here for mesh tying in the presence of gaps and overlaps. The least-squares formulation for transmission problems [5] is extended to settings where subdomain boundaries are not spatially coincident. The new method is consistent in the sense that it recovers exactly global polynomial solutions that are in the finite element space. As a result, the least-squares mesh tying method passes a patch test of the order of the finite element space by construction. This attractive computational property is illustrated by numerical experiments.