Second-order $L^{1}$ convergence of the finite element approximation

Let $H=L^{1}([x_{\min},x_{\max}],\mathbb{R}_{+})$, let $H^{N}$ be the piecewise-constant approximation space generated by the basis functions $\beta_i^N$, and let $f^{N}(t,x)$ solve the projected finite-dimensional approximation of the Smoluchowski coagulation equation. For time nodes $t_k$ and spatial domain $\boldsymbol{X}$, second-order convergence conjecture. The approximate solution, $f^{N}(t_{k},x)$, converges to the analytical solution with order $2$ in $L^{1}[\boldsymbol{X}]$. This conjecture is motivated by numerical experiments and reported superconvergence results for discontinuous Galerkin methods; an analytical investigation is left for future work.