Superconvergence of the local discontinuous galerkin method applied to the one-dimensional second-order wave equation

We analyze the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the second-order wave equation in one space dimension. With a suitable projection of the initial conditions for the LDG scheme, we prove that the LDG solution and its spatial derivative are O (h p + 3 / 2) super close to particular projections of the exact solutions for pth-degree polynomial spaces. We use these results to show that the significant parts of the discretization errors for the LDG solution and its derivative are proportional to (p + 1) -degree right and left Radau polynomials, respectively. These results allow us to prove that the p-degree LDG solution and its derivative are O (h p + 3 / 2) superconvergent at the roots of (p + 1) -degree right and left Radau polynomials, respectively, while computational results show higher O (h p + 2) convergence rate. Superconvergence results can be used to construct asymptotically correct a posteriori error estimates by solving a local steady problem on each element. This will be discussed further in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivative converge to the true spatial errors in the L ²-norm under mesh refinement.