Analysis of optimal superconvergence of the local discontinuous Galerkin method for nonlinear fourth-order boundary value problems

Abstract

This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value problems of the type $u^{(4)}=f(x,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime })$ , x ∈ [a,b] with classical boundary conditions at the endpoints. Convergence properties for the solution and for all three auxiliary variables are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th-order convergence, when polynomials of degree p are used. We also prove that the derivatives of the errors between the LDG solutions and Gauss-Radau projections of the exact solutions in the L2 norm are superconvergent with order p + 1. Furthermore, a (2p + 1)th-order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages is also obtained under quasi-uniform meshes. Finally, we prove that the LDG solutions are superconvergent with an order of p + 2 toward particular projections of the exact solutions. The error analysis presented in this paper is valid for p ≥ 1. Numerical experiments indicate that our theoretical findings are optimal.