Exact-curved Lagrange finite elements for the Poisson problem in two dimensions

We develop an exact-curved Lagrange finite element framework for the Poisson problem on two-dimensional curved domains. The element map is factorised as $ F_K=\Psi_K\circ\Phi_{T_K}$, where $\Phi_{T_K}$ maps the reference triangle to an affine core and $\Psi_K$ maps the affine core to the physical curved element. This factorisation separates affine scaling from curvature effects and allows the interpolation analysis to be carried out first on the affine core and then transferred to the exact curved element. For conforming linear Lagrange elements, we prove local $L^2$- and $H^1$-interpolation estimates on exact curved triangles. The estimates are expressed in terms of transported directional derivatives on the physical element, and the constants are independent of the anisotropic shape of the affine core under the stated semi-regularity assumptions. These interpolation estimates are then applied to derive energy-norm and $L^2$-error estimates for the Poisson problem. Numerical results on the unit disk illustrate the difference between straight-sided and curved geometric representations: the curved geometry reduces the geometric error substantially, while the leading finite element error remains governed by the $\mathbb{P}^1$ approximation.