TY - JOUR
T1 - An Optimal a Posteriori Error Estimates of the Local Discontinuous Galerkin Method for the Second-Order Wave Equation in One Space Dimension
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 355
EP - 380
PY - 2017
DA - 2017/06
SN - 14
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/10012.html
KW - Local discontinuous Galerkin method, second-order wave equation, superconvergence, Radau points, a posteriori error estimation.
AB - <p style="text-align: justify;">In this paper, we provide the optimal convergence rate of a posteriori error estimates
for the local discontinuous Galerkin (LDG) method for the second-order wave equation in one
space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence
result in [W. Cao, D. Li and Z. Zhang, Commun. Comput. Phys. 21 (1) (2017) 211-236]. We
first prove that the LDG solution and its spatial derivative, respectively, converge in the $L^2$-norm
to $(p+1)$-degree right and left Radau interpolating polynomials under mesh refinement. The
order of convergence is proved to be $p+2$, when piecewise polynomials of degree at most $p$ are
used. We use these results to show that the leading error terms on each element for the solution
and its derivative are proportional to $(p+1)$-degree right and left Radau polynomials. These
new results enable us to construct residual-based a posteriori error estimates of the spatial errors.
We further prove that, for smooth solutions, these a posteriori LDG error estimates converge, at
a fixed time, to the true spatial errors in the $L^2$-norm at $\mathcal{O}(h^{p+2})$ rate. Finally, we show that
the global effectivity indices in the $L^2$-norm converge to unity at $\mathcal{O}(h)$ rate. The current results
improve upon our previously published work in which the order of convergence for the a posteriori
error estimates and the global effectivity index are proved to be $p+3/2$ and $1/2$, respectively. Our
proofs are valid for arbitrary regular meshes using $P^p$ polynomials with $p\geq1$. Several numerical
experiments are performed to validate the theoretical results.</p>