Optimal $L^{\mathrm{\infty }}$-error estimate for isoparametric finite element method in a smooth domain

We consider the isoparametric finite element method (FEM) for the Poisson equation in a smooth domain with the homogeneous Dirichlet boundary condition. Because the boundary is curved, standard triangulated meshes do not exactly fit it. Thereby we need to introduce curved elements if better accuracy than linear FEM is desired, which necessitates the use of isoparametric FEMs. We establish optimal rate of convergence $O(h^{k+1})$ in the $L^{\mathrm{\infty }}$-norm for $k\ge 2$, by extending the approach of our previous work [Kashiwabara and Kemmochi, Numer.\ Math.\ $\mathbf{\text{144}}$, 553–584 (2020)] developed for Neumann boundary conditions and $k=1$.