  In this paper the authors study a non-linear elliptic-parabolic system, which
is motivated by mathematical models for lithium-ion batteries. One state
satisfies a parabolic reaction diffusion equation and the other one an elliptic
equation. The goal is to determine several scalar parameters in the coupled
model in an optimal manner by utilizing a reliable reduced-order approach based
on the reduced basis (RB) method. However, the states are coupled through a
strongly non-linear function, and this makes the evaluation of online-efficient
error estimates difficult. First the well-posedness of the system is proved.
Then a Galerkin finite element and RB discretization are described for the
coupled system. To certify the RB scheme hierarchical a-posteriori error
estimators are utilized in an adaptive trust-region optimization method.
Numerical experiments illustrate good approximation properties and efficiencies
by using only a relatively small number of reduced basis functions.
