A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

In this paper, we propose and analyze a new superconvergent local discontinuous Galerkin (LDG) method equipped with an element residual error estimator for the spatial discretization of the second-order wave equation on Cartesian grids. We prove the L² stability, the energy conserving property, and optimal L² error estimates for the semi-discrete formulation. In particular, we identify special numerical fluxes for which the L²-norm of the solution and its gradient are of order p+1, when tensor product polynomials of degree at most p are used. We further perform a local error analysis and show that the leading term of the LDG error is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, the LDG solution is O(h^p+2) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree right Radau polynomial. Furthermore, numerical computations show that the first component of the solution's gradient is O(h^p+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h^p+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Computational results indicate that global superconvergence holds for LDG solutions. We use the superconvergence results to construct a posteriori LDG error estimates. These error estimates are computationally simple and are obtained by solving local steady problems with no boundary condition on each element. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.