A Superconvergent Local Discontinuous Galerkin Method for Nonlinear Fourth-Order Boundary-Value Problems

Abstract

In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form [Formula: see text]. We prove optimal [Formula: see text] error estimates for the solution and for the three auxiliary variables that approximate the first, second, and third-order derivatives. The order of convergence is proved to be [Formula: see text], when piecewise polynomials of degree at most [Formula: see text] are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order [Formula: see text] toward the derivatives of Gauss–Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order [Formula: see text] toward Gauss–Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is [Formula: see text]. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree [Formula: see text] and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.