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

In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form u(4) + f(x,u) = 0. We prove optimal L2 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 p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order p + 1 toward the derivatives of Gauss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is p + 2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥ 1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.