The numerical study of a microscale model for lithium-ion batteries

We consider a thermodynamic consistent microscale model from for the transport processes of lithium ion concentration and charge in lithium ion batteries. A fully convergent finite element discretisation in space of arbitrary order with uniform time stepping is discussed for this strongly nonlinear coupled system of nonlinear PDEs. We analyse the error for higher order finite element methods on a representative geometry, which shows already the difficulties when dealing with a realistic microstructure. Moreover we derive explicit analytical formulas for the exact solutions for the elliptic subproblem and the complete time-dependent problem for a non-trivial geometry. These results show the need for appropriate boundary approximations as well as adaptive refinement strategies. In addition we incorporate a two-phase intercalation model for the electrode. This moving boundary model, also called Stefan problem, has also been implemented. Simulations of our battery model including the two-phase model will be shown.