An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems

In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form $u^{\prime \prime }=f(x,u),\ x\in [a,b]$ subject to some suitable boundary conditions at the endpoint of the interval [a, b]. We first use the superconvergence results proved in the first part of this paper as reported by Baccouch (Numer. Algorithm. 79(3), 697–718 2018) to show that the significant parts of the local discretization errors are proportional to (p + 1)-degree Radau polynomials, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed a posteriori error estimates converge to the exact errors in the L2-norm with order of convergence p + 3/2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p ≥ 1. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.