A local discontinuous Galerkin method for the second-order wave equation

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

“…We used these results to construct asymptotically correct a posteriori error estimates by solving local steady problem with no boundary conditions on each element. This paper is a natural continuation of the work done in [22] by the author where the first superconvergence results and a posteriori error estimates of the semi-discrete LDG method applied to the onedimensional second-order wave equation were investigated. In [58], we analyzed the LDG method introduced by the author in [22] for solving the one-dimensional second-order wave equation.…”

“…This paper is a natural continuation of the work done in [22] by the author where the first superconvergence results and a posteriori error estimates of the semi-discrete LDG method applied to the onedimensional second-order wave equation were investigated. In [58], we analyzed the LDG method introduced by the author in [22] for solving the one-dimensional second-order wave equation. We used a suitable projection of the initial conditions for the numerical scheme and proved optimal L 2 error estimates for the LDG solution and its spatial derivative.…”

“…In [22], we presented new superconvergence results for the semi-discrete LDG method applied to the second-order scalar wave equation in one space dimension. We performed an error analysis on one element and showed that the p-degree LDG solution and its spatial derivative are O(h p+2 ) superconvergent at the roots of (p + 1)-degree right and left Radau polynomials, respectively.…”

“…For an introduction to the subject of a posteriori error estimation see the monograph of Ainsworth and Oden [19]. Superconvergence properties for finite element and DG methods have been studied in [3,5,20,2] for ordinary differential equations, [21,22,3,23] for hyperbolic problems and [24][25][26]23,[27][28][29][30][31] for diffusion and convection-diffusion problems. Several a posteriori DG error estimates are known for hyperbolic [32][33][34] and diffusive [35,36] problems.…”

Superconvergence of discontinuous Galerkin solutions for higher-order ordinary differential equations

“…Yang and Shu [15] applied the LDG to one-dimensional linear parabolic equation, they proved that the error between the LDG solution and the exact solution is (p + 2)th order superconvergent at the Radau points with suitable initial discretization and they proved that the LDG solution is (p + 2)th order superconvergent for the error to a particular projection of the exact solution when using piecewise pth degree polynomials. Baccouch analyzed the superconvergence properties of the LDG method applied to the second-order wave equation in one space dimension, he [4] showed that the LDG solution is O (h p+2 ) superconvergent at the (p + 1)-degree right-Radau polynomial and the solution's derivative is O (h p+2 ) superconvergent at the (p + 1)-degree left-Radau polynomial; and he [7] proved 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. Unfortunately, researchers were not fully satisfied with the LDG method due to the introduction of new auxiliary variables and transformation of the original equation into a system of several first order equations which leads to a more complex DG method with expensive computational cost.…”