Optimal convergence in finite element semidiscrete error analysis of the Doyle–Fuller–Newman model beyond one dimension with a novel projection operator

We present a finite element semidiscrete error analysis for the Doyle–Fuller–Newman model, which is the most popular model for lithium-ion batteries. Central to our approach is a novel projection operator designed for the pseudo-($N$+1)-dimensional equation, offering a powerful tool for multiscale equation analysis. Our results bridge a gap in the analysis for dimensions $2 \le N \le 3$ and achieve optimal convergence rates of $h+(\varDelta r)^{2}$. Additionally, we perform a detailed numerical verification, marking the first such validation in this context. By avoiding the change of variables our error analysis can also be extended beyond isothermal conditions.