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

Adaptive Parameter Optimization For An Elliptic-Parabolic System Using The Reduced-Basis Method With Hierarchical A-Posteriori Error Analysis