<p>A simultaneous identification problem is investigated for two time independent coefficients in a nonlinear phase field system. This coefficient identification problem is formulated as an optimal control problem governed by the phase field equations, for which an equivalent optimality system is derived using the optimal control theory. A finite element approximation scheme is established, in which the state, co-state, and control variables are all discretized using piecewise linear continuous functions. A priori error estimates for these variables are rigorously derived. Due to the mutual independence of the two coefficients, an iterative algorithm is constructed to solve them via two corresponding elliptic boundary value problems. Finally, numerical experiments are conducted to validate the theoretical results.</p>