Optimal constrained interpolation in mesh-adaptive finite element modelling

Mesh-to-mesh Galerkin $L^2$ projection allows piecewise polynomial unstructured finite element data to be interpolated between two nonmatching unstructured meshes of the same domain. The interpolation is by definition optimal in an $L^2$ sense, and subject to fairly weak assumptions conserves the integral of an interpolated function. However other properties, such as the $L^2$ norm, or the weak divergence of a vector-valued function, can still be adversely affected by the interpolation. This is an important issue for calculations in which numerical dissipation should be minimized, or for simulations of incompressible flow. This paper considers extensions to mesh-to-mesh Galerkin $L^2$ projection which are $L^2$ optimal and ensure exact conservation of key discrete properties, including preservation of both the $L^2$ norm and the integral, and preservation of both the $L^2$ norm and weak incompressibility. The accuracy of the interpolants is studied. The utility of the interpolants is studied via adaptive mesh simulations of the two-dimensional lock-exchange problem, which are simulated using a combination of Fluidity and the FEniCS system.