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OPTIMAL A POSTERIORI ERROR ESTIMATES OF THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR CONVECTIONDIFFUSION PROBLEMS IN ONE SPACE DIMENSION

Mahboub Baccouch   

  1. Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182
  • Received:2015-06-23 Revised:2015-12-28 Online:2016-09-15 Published:2016-09-15

Mahboub Baccouch. OPTIMAL A POSTERIORI ERROR ESTIMATES OF THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR CONVECTIONDIFFUSION PROBLEMS IN ONE SPACE DIMENSION[J]. Journal of Computational Mathematics, 2016, 34(5): 511-531.

In this paper, we derive optimal order a posteriori error estimates for the local discontinuous Galerkin (LDG) method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconvergence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 (2015), pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-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. These results are used 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. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at O(hp+2) rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h) rate. These 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 Pp polynomials with p≥1. Several numerical experiments are performed to validate the theoretical results.

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[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.

[2] S. Adjerid, M. Baccouch, A superconvergent local discontinuous Galerkin method for elliptic problems, Journal of Scientific Computing, 52(2012), 113-152.

[3] S. Adjerid, D. Issaev, Superconvergence of the local discontinuous Galerkin method applied to diffusion problems, in: K. Bathe (ed.), Proceedings of the Third MIT Conference on Computational Fluid and Solid Mechanics, vol. 3, Elsevier, 2005.

[4] S. Adjerid, A. Klauser, Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems, Journal of Scientific Computing, 22(2005), 5-24.

[5] M. Ainsworth, J.T. Oden, A posteriori Error Estimation in Finite Element Analysis, John Wiley, New York, 2000.

[6] M. Baccouch, A local discontinuous Galerkin method for the second-order wave equation, Computer Methods in Applied Mechanics and Engineering, 209-212(2012), 129-143.

[7] M. Baccouch, Asymptotically exact a posteriori LDG error estimates for one-dimensional transient convection-diffusion problems, Applied Mathematic and Computation, 226(2014), 455-483.

[8] M. Baccouch, The local discontinuous Galerkin method for the fourth-order Euler-Bernoulli partial differential equation in one space dimension. Part I: Superconvergence error analysis, Journal of Scientific Computing, 59(2014), 795-840.

[9] M. Baccouch, Superconvergence and a posteriori error estimates for the LDG method for convection-diffusion problems in one space dimension, Computers & Mathematics with Applications, 67(2014), 1130-1153.

[10] M. Baccouch, A superconvergent local discontinuous Galerkin method for the second-order wave equation on cartesian grids, Computers and Mathematics with Applications, 68(2014), 1250-1278.

[11] M. Baccouch, Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation, Numerical methods of partial differential equations, 31(2015), 1461-1491.

[12] M. Baccouch, S. Adjerid, A posteriori local discontinuous Galerkin error estimation for twodimensional convection-diffusion problems, Journal of Scientific Computing, 62(2014), 399-430.

[13] W. Cao, Z. Zhang, Superconvergence of local discontinuous Galerkin methods for one-dimensional linear parabolic equations, Mathematics of Computation. Article electronically published on June 1, 2015, DOI: http://dx.doi.org/10.1090/mcom/2975(toappearinprint).

[14] P. Castillo, A superconvergence result for discontinuous Galerkin methods applied to elliptic problems, Computer Methods in Applied Mechanics and Engineering, 192(2003), 4675-4685.

[15] P. Castillo, A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems, Applied Numerical Mathematics, 56(2006), 1307-1313.

[16] P. Castillo, B. Cockburn, D. Schötzau, C. Schwab, Optimal a priori error estimates for the hpversion of the local discontinuous Galerkin method for convection-diffusion problems, Mathematic of Computation, 71(2002), 455-478.

[17] F. Celiker, B. Cockburn, Superconvergence of the numerical traces for discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Mathematics of Computation, 76(2007), 67-96.

[18] Y. Cheng, C.W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47(2010), 4044-4072.

[19] P.G. Ciarlet, The finite element method for elliptic problems, North-Holland Pub. Co., Amsterdam-New York-Oxford, 1978.

[20] B. Cockburn, A simple introduction to error estimation for nonlinear hyperbolic conservation laws, in: Proceedings of the 1998 EPSRC Summer School in Numerical Analysis, SSCM, volume 26 of the Graduate Student's Guide for Numerical Analysis, pages 1-46, Springer, Berlin, 1999.

[21] B. Cockburn, P.A. Gremaud, Error estimates for finite element methods for nonlinear conservation laws, SIAM Journal on Numerical Analysis, 33(1996) 522-554.

[22] B. Cockburn, G. Kanschat, D. Schötzau, A locally conservative LDG method for the incompress-ible Navier-Stokes equations, Mathematics of compuatation, 74(2004), 1067-1095.

[23] B. Cockburn, G. Kanschat, D. Schötzau, The local discontinuous Galerkin method for linearized incompressible fluid flow: a review, Computers & Fluids 34(4-5) (2005), 491-506, residual Distribution Schemes, Discontinuous Galerkin Schemes and Adaptation.

[24] B. Cockburn, G.E. Karniadakis, C.W. Shu, Discontinuous Galerkin Methods Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, 2000.

[25] B. Cockburn, C.W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws II: General framework, Mathematics of Computation, 52(1989), 411-435.

[26] B. Cockburn, C.W. Shu, The local discontinuous Galerkin method for time-dependent convectiondiffusion systems, SIAM Journal on Numerical Analysis, 35(1998), 2440-2463.

[27] K.D. Devine, J.E. Flaherty, Parallel adaptive hp-refinement techniques for conservation laws, Computer Methods in Applied Mechanics and Engineering, 20(1996), 367-386.

[28] J.E. Flaherty, R. Loy, M.S. Shephard, B. K. Szymanski, J. D. Teresco, L. H. Ziantz, Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws, Journal of Parallel and Distributed Computing, 47(1997), 139-152.

[29] S. Gottlieb, C.W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43:1(2001), 89-112.

[30] R. Hartmann, P. Houston, Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservations laws, SIAM Journal on Scientific Computing, 24(2002), 979-1004.

[31] P. Houston, D. Schötzau, T. Wihler, Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Mathematical Models and Methods in Applied Sciences, 17(2007), 33-62.

[32] X. Meng, C.-W. Shu, B. Wu, Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension, IMA Journal of Numerical Analysis, 32(2012), 1294-1328.

[33] X. Meng, C.W. Shu, Q. Zhang, B. Wu, Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension, SIAM Journal on Numerical Analysis, 50:5(2012), 2336-2356.

[34] T.E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM Journal on Numerical Analysis, 28:1(1991), 133-140.

[35] W.H. Reed, T.R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos (1973).

[36] B. Rivière, M. Wheeler, A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems, Computational and Applied Mathematics, 46(2003), 143-163.

[37] D. Schötzau, C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM Journal on Numerical Analysis, 38(2000), 837-875.

[38] Y. Yang, C.W. Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations, SIAM Journal on Numerical Analysis, 50(2012), 3110-3133.

[39] Y. Yang, C.W. Shu, Analysis of sharp superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations, Journal of Computational Mathematics, 33(2015), 323-340.
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