TY - JOUR

T1 - A local discontinuous Galerkin method for the second-order wave equation

AU - Baccouch, Mahboub

N1 - Funding Information:
This research was partially supported by the NASA Nebraska Space Grant Program at the University of Nebraska at Omaha. The author also acknowledge the University Committee on Research and Creative Activity (UCRCA) at the University of Nebraska at Omaha, for continuous funding support to this research. The author would also like to thank the two referees for their constructive comments and remarks which helped improve the quality and readability of the paper.

PY - 2012/2/1

Y1 - 2012/2/1

N2 - In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show O(hp+1)L2 convergence rate for the LDG solution and O(h p+2) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p+1)-degree right Radau and (p+1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree right Radau polynomial and the derivative of the LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement.

AB - In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show O(hp+1)L2 convergence rate for the LDG solution and O(h p+2) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p+1)-degree right Radau and (p+1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree right Radau polynomial and the derivative of the LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement.

KW - A posteriori error estimation

KW - Local discontinuous Galerkin method

KW - Second-order wave equation

KW - Superconvergence

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U2 - 10.1016/j.cma.2011.10.012

DO - 10.1016/j.cma.2011.10.012

M3 - Article

AN - SCOPUS:83455262231

VL - 209-212

SP - 129

EP - 143

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

ER -