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

[摘要] 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.

[学科分类] 应用数学

[关键词] discontinuous Galerkin method;advection-reaction equation;optimal convergence rate;a posteriori error estimate