A gradient based resolution strategy for a PDE-constrained optimization approach for 3D-1D coupled problems

Coupled 3D-1D problems arise in many practical applications, an attempt to reduce the computational burden simulations where cylindrical inclusions with a small section are embedded much larger domain. Nonetheless resolution of such can be non trivial, both from mathematical and geometrical standpoint. Indeed coupling requires operate standard function spaces, and, also, simulation geometries complex for presence multiple intersecting domains. Recently, PDE-constrained optimization based formulation has been proposed problems, proving well posed allowing use conforming meshes discrete problem. Here unconstrained problem is derived efficient gradient solver formulation, along suitable preconditioner speed up iterative solver. Some numerical tests on quite configurations discussed show viability method .