Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations

Tie Zhu Zhang; Shu Hua Zhang

Applications of Mathematics (2015)

  • Volume: 60, Issue: 1, page 1-20
  • ISSN: 0862-7940

Abstract

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We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in d space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order k + 1 in the L 2 -norm if the method uses polynomials of order k . Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order k + 1 . Further we consider a residual-based a posteriori error estimate and give the global upper bound and local lower bound on the error in the DG-norm, which is stronger than the L 2 -norm. The key elements in our a posteriori analysis are the saturation assumption and an interpolation estimate between the DG spaces. We show that the a posteriori error bounds are efficient and reliable. Finally, some numerical experiments are presented to illustrate the theoretical analysis.

How to cite

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Zhang, Tie Zhu, and Zhang, Shu Hua. "Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations." Applications of Mathematics 60.1 (2015): 1-20. <http://eudml.org/doc/262134>.

@article{Zhang2015,
abstract = {We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in $d$ space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order $k+1$ in the $L_2$-norm if the method uses polynomials of order $k$. Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order $k+1$. Further we consider a residual-based a posteriori error estimate and give the global upper bound and local lower bound on the error in the DG-norm, which is stronger than the $L_2$-norm. The key elements in our a posteriori analysis are the saturation assumption and an interpolation estimate between the DG spaces. We show that the a posteriori error bounds are efficient and reliable. Finally, some numerical experiments are presented to illustrate the theoretical analysis.},
author = {Zhang, Tie Zhu, Zhang, Shu Hua},
journal = {Applications of Mathematics},
keywords = {discontinuous Galerkin method; advection-reaction equation; optimal convergence rate; a posteriori error estimate; discontinuous Galerkin method; advection-reaction equation; optimal convergence rate; a posteriori error estimate; numerical experiments},
language = {eng},
number = {1},
pages = {1-20},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations},
url = {http://eudml.org/doc/262134},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Zhang, Tie Zhu
AU - Zhang, Shu Hua
TI - Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 1
EP - 20
AB - We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in $d$ space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order $k+1$ in the $L_2$-norm if the method uses polynomials of order $k$. Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order $k+1$. Further we consider a residual-based a posteriori error estimate and give the global upper bound and local lower bound on the error in the DG-norm, which is stronger than the $L_2$-norm. The key elements in our a posteriori analysis are the saturation assumption and an interpolation estimate between the DG spaces. We show that the a posteriori error bounds are efficient and reliable. Finally, some numerical experiments are presented to illustrate the theoretical analysis.
LA - eng
KW - discontinuous Galerkin method; advection-reaction equation; optimal convergence rate; a posteriori error estimate; discontinuous Galerkin method; advection-reaction equation; optimal convergence rate; a posteriori error estimate; numerical experiments
UR - http://eudml.org/doc/262134
ER -

References

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