Geodesic
Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations
Applications of Mathematics, Tome 60 (2015) no. 1, pp. 1-20

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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.
DOI : 10.1007/s10492-015-0082-x
Classification : 65M60, 65N15, 65N30
Keywords: discontinuous Galerkin method; advection-reaction equation; optimal convergence rate; a posteriori error estimate 
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Zhang, Tie; Zhang, Shuhua. Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations. Applications of Mathematics, Tome 60 (2015) no. 1, pp. 1-20. doi: 10.1007/s10492-015-0082-x

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