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

Abstract

Conforming mapped Lagrange finite elements of any fixed degree are analysed for the Poisson problem on exact curved triangulations in two dimensions. The element map is factorised into an anisotropic affine core map and a smooth curved correction. Assuming compatible edge parametrisations, semi-regular affine cores and uniform bounds on the curved corrections, we obtain anisotropic $L^2$- and $H^1$-interpolation estimates with constants independent of the aspect ratios of the cores. The estimates retain lower-order terms induced by the non-affine maps, and these terms cannot be omitted in general. For sufficiently regular solutions, the Galerkin errors on the exact domain converge with the optimal orders in the energy norm and, on convex domains, in the $L^2$ norm. A fixed smooth diffeomorphism maps a semi-regular family of polygonal triangulations to exact curved triangulations satisfying these assumptions, even when the core aspect ratios are unbounded. An explicit family with an exact circular boundary arc is given. Numerical experiments with linear and quadratic elements confirm the predicted rates. At a fixed horizontal mesh width, the errors remain bounded as the aspect ratio increases.