The Fully Constrained Formulation: local uniqueness and numerical accuracy

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In this talk I will introduce the Fully Constrained Formulation (FCF) of General Relativity. In this formulation one has a hyperbolic sector and an elliptic one. The constraint equations are solved in each time step and are encoded in the elliptic sector; this set of equations have to be solved to compute initial data even if a free evolution scheme is used for a posterior dynamical evolution. Other formulations (like the XCTS formulation) share a similar elliptic sector. I will comment about the local uniqueness issue of the elliptic sector in the FCF. I will also described briefly the hyperbolic sector. I will finish with some recent reformulation of the equations which keeps the good properties of the local uniqueness, improves the numerical accuracy of the system and gives some additional information.