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 is 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 bases functions.

翻译：本文研究了一类非线性椭圆-抛物系统，该系统源于锂离子电池的数学模型。其中一个状态变量满足抛物型反应扩散方程，另一个状态变量满足椭圆型方程。研究目标是通过基于约化基（RB）方法的可靠降阶手段，以最优方式确定耦合模型中的多个标量参数。然而，状态变量通过强非线性函数耦合，这为在线高效误差估计的评估带来了困难。本文首先证明了系统的适定性，随后描述了耦合系统的伽辽金有限元与RB离散化方案。为验证RB方案的可靠性，在自适应信赖域优化方法中采用了分层后验误差估计器。数值实验表明，仅使用少量约化基函数即可获得良好的逼近性质与计算效率。