A posteriori error estimate for a PDE-constrained optimization formulation for the flow in DFNs

Flows in fractured media have been modeled with many different approaches in order to get reliable and efficient simulations for many critical applications. The common issues to be tackled are the wide range of scales involved in the phenomenon, the complexity of the domain, and the huge computational cost. In this paper we introduce residual-based “a posteriori” error estimates for a formulation of the flow in the discrete fracture networks based on a constrained optimization approach (see Berrone, Pieraccini, and Scialo` [SIAM J. Sci. Comput., 35 (2013), pp. B487–B510], [SIAM J. Sci. Comput., 35 (2013), pp. A908–A935], [J. Comput. Phys., 256 (2014), pp. 838–853]), suitable to overcome all the difficulties related to a good quality mesh generation with conformity requirement.