(untitled) (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 Galerkin methods and a C^2-C^0 Petrov-Galerkin method. Existence, uniqueness and optimal error estimates of the numerical solution are established. A unified approach is proposed to study the superconvergence property of these methods. We prove that, for kth-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 superconvergence points for the l-th (0≤l≤m+1) derivate approximations are also discovered, which are identified as roots of special Jacobi polynomials, Lobatto points, and Gauss points. As a by-product, we prove that the C^m finite element solution is superconvergent towards a particular Jacobi projection of the exact solution in the H^l (0≤ l ≤ m+1) norms. All theoretical findings are confirmed by numerical experiments.