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 L2L2 stability, the energy conserving property, and optimal L2L2 error estimates for the semi-discrete formulation. In particular, we identify special numerical fluxes for which theL2L2-norm of the solution and its gradient are of order p+1p+1, when tensor product polynomials of degree at most pp 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)(p+1)-degree right Radau polynomials in the xx and yy directions. Thus, the LDG solution is O(hp+2)O(hp+2) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)(p+1)-degree right Radau polynomial. Furthermore, numerical computations show that the first component of the solution’s gradient is O(hp+2)O(hp+2) superconvergent at tensor product of the roots of left Radau polynomial in xx and right Radau polynomial in yy while the second component is O(hp+2)O(hp+2) superconvergent at the tensor product of the roots of the right Radau polynomial in xx and left Radau polynomial in yy. 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.