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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
Summary:
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.
References:
[1] Achchab, B., Achchab, S., Agouzal, A.: Some remarks about the hierarchical a posteriori error estimate. Numer. Methods Partial Differ. Equations 20 (2004), 919-932. DOI 10.1002/num.20016 | MR 2092413 | Zbl 1059.65097
[2] Becker, R., Hansbo, P., Stenberg, R.: A finite element method for domain decomposition with non-matching grids. M2AN, Math. Model. Numer. Anal. 37 (2003), 209-225. DOI 10.1051/m2an:2003023 | MR 1991197 | Zbl 1047.65099
[3] Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996), 2431-2444. DOI 10.1137/S0036142994264079 | MR 1427472 | Zbl 0866.65071
[4] Burman, E.: A posteriori error estimation for interior penalty finite element approximations of the advection-reaction equation. SIAM J. Numer. Anal. 47 (2009), 3584-3607. DOI 10.1137/080733899 | MR 2576512 | Zbl 1205.65249
[5] Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 193 (2004), 1437-1453. DOI 10.1016/j.cma.2003.12.032 | MR 2068903 | Zbl 1085.76033
[6] Cairlet, P. G.: The Finite Element Methods for Elliptic Problems. Repr., unabridged republ. of the orig. 1978. Classics in Applied Mathematics 40 SIAM, Philadelphia (2002). MR 1930132
[7] Cockburn, B., Dong, B., Guzmán, J.: Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46 (2008), 1250-1265. DOI 10.1137/060677215 | MR 2390992 | Zbl 1168.65058
[8] Dörfler, W., Nochetto, R. H.: Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002), 1-12. DOI 10.1007/s002110100321 | MR 1896084 | Zbl 0995.65109
[9] Eriksson, K., Johnson, C.: Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput. 60 (1993), 167-188. DOI 10.1090/S0025-5718-1993-1149289-9 | MR 1149289 | Zbl 0795.65074
[10] Ern, A., Guermond, J.-L.: Discontinuous Galerkin methods for Friedrichs' systems. I. General Theory. SIAM J. Numer. Anal. 44 (2006), 753-778. DOI 10.1137/050624133 | MR 2218968 | Zbl 1122.65111
[11] Friedrichs, K. O.: Symmetric positive linear differential equations. Commun. Pure Appl. Math. 11 (1958), 333-418. DOI 10.1002/cpa.3160110306 | MR 0100718 | Zbl 0083.31802
[12] Houston, P., Rannacher, R., Süli, E.: A posteriori error analysis for stabilised finite element approximations of transport problems. Comput. Methods Appl. Mech. Eng. 190 (2000), 1483-1508. DOI 10.1016/S0045-7825(00)00174-2 | MR 1807010 | Zbl 0970.65115
[13] Houston, P., Süli, E.: $hp$-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems. SIAM J. Sci. Comput. 23 (2001), 1226-1252. DOI 10.1137/S1064827500378799 | MR 1885599 | Zbl 1029.65130
[14] Johnson, C., Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46 (1986), 1-26. DOI 10.1090/S0025-5718-1986-0815828-4 | MR 0815828 | Zbl 0618.65105
[15] Lasaint, P., Raviart, P. A.: On a finite element method for solving the neutron transport equation. Proc. Symp. Math. Aspects Finite Elem. Partial Differ. Equat., Madison 1974 Academic Press New York (1974), 89-123. MR 0658142 | Zbl 0341.65076
[16] Peterson, T. E.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28 (1991), 133-140. DOI 10.1137/0728006 | MR 1083327 | Zbl 0729.65085
[17] Reed, W. H., Hill, T. R.: Triangular mesh methods for the neutron transport equation. Tech. Report LA-Ur-73-479, Los Alamos Scientific Laboratory, 1973.
[18] Richter, G. R.: An optimal-order error estimate for the discontinuous Galerkin method. Math. Comput. 50 (1988), 75-88. DOI 10.1090/S0025-5718-1988-0917819-3 | MR 0917819 | Zbl 0643.65059
[19] Richter, G. R.: On the order of convergence of the discontinuous Galerkin method for hyperbolic equations. Math. Comput. 77 (2008), 1871-1885. DOI 10.1090/S0025-5718-08-02126-1 | MR 2429867 | Zbl 1198.65196
[20] Verfürth, R.: A posteriori error estimators for convection-diffusion equations. Numer. Math. 80 (1998), 641-663. DOI 10.1007/s002110050381 | MR 1650051 | Zbl 0913.65095
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