Optimal-rate error estimates and a twice decoupled solver for a backward Euler finite element scheme of the Doyle–Fuller–Newman model of lithium-ion cells

We investigate the convergence of a backward Euler finite element discretization applied to a multi-domain and multi-scale elliptic–parabolic system, derived from the Doyle-Fuller-Newman model for lithium-ion cells. We establish optimal-order error estimates for the solution in the norms $l^2\left(H^1\right)$ and $l^2\left(L^2\left(H_r^q\right)\right)$, $q=0,1$. To improve computational efficiency, we propose a novel solver that accelerates the solution process and controls memory usage. Numerical experiments with realistic battery parameters validate the theoretical error rates and demonstrate the significantly superior performance of the proposed solver over existing solvers.