Superconvergence Analysis of $C^m$ Finite Element Methods for Fourth-Order Elliptic Equations I: One Dimensional Case (Communications in Computational Physics, 2023)
In this paper, we study three families of $C^m (m=0,1,2)$ finite element methods for one dimensional fourth-order equations. They include $C^0$ and $C^1$ Galerkinmethods and a $C^2-C^0$ Petrov-Galerkin method. Existence, uniqueness and optimalerror estimates of the numerical solution are established. A unified approach is proposed to study the superconvergence property of these methods. We prove that, for $k$th-order elements, the $C^0$ and $C^1$ finite element solutions and their derivative are superconvergent with rate $h^{2k−2} (k≥3)$ at all mesh nodes; while the solution of the $C^2-C^0$ Petrov-Galerkin method and its first- and second-order derivatives are superconvergent with rate $h^{2k−4} (k≥5)$ at all mesh nodes. Furthermore, interior superconvergencepoints for the $l$-${\rm th} (0≤l≤m+1)$ derivate approximations are also discovered, which areidentified as roots of special Jacobi polynomials, Lobatto points, and Gauss points. Asa by-product, we prove that the $C^m$ finite element solution is superconvergent towardsa particular Jacobi projection of the exact solution in the $H^l(0≤ l ≤ m+1)$ norms. Alltheoretical findings are confirmed by numerical experiments.
Keywords: $C^m$ finite element methods; superconvergence; fourth-order elliptic equations.