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July  2013, 18(5): 1493-1505. doi: 10.3934/dcdsb.2013.18.1493

$L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems

Huiqing Zhu 1, and  Runchang Lin 2,

1. 

Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406, United States
2. 

Department of Engineering, Mathematics, and Physics, Texas A&M International University, Laredo, TX 78041, United States

Received  April 2012 Revised  October 2012 Published  March 2013

Pointwise error estimates of the local discontinuous Galerkin (LDG) method for a one-dimensional singularly perturbed problem are studied. Several uniform $L^\infty$ error bounds for the LDG approximation to the solution and its derivative are established on a Shishkin-type mesh. Numerical experiments are presented.
Keywords: Shishkin mesh., singular perturbation, Local discontinuous Galerkin method.
Mathematics Subject Classification: Primary: 65L10, 65L11; Secondary: 65L60, 65L2.
Citation: Huiqing Zhu, Runchang Lin. $L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1493-1505. doi: 10.3934/dcdsb.2013.18.1493
References:
[1]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.  Google Scholar

[2]

P. Castillo, B. Cockburn, D. Sch$\ddoto$tzau and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Math. Comp., 71 (2002), 455-478. doi: 10.1090/S0025-5718-01-01317-5.  Google Scholar

[3]

F. Celiker and B. Cockburn, Superconvergence of the numerical traces of discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Math. Comp., 76 (2007), 67-96. doi: 10.1090/S0025-5718-06-01895-3.  Google Scholar

[4]

F. Celiker, Z. Zhang and H. Zhu, Nodal superconvergence of SDFEM for singularly perturbed problems, J. Sci. Comput., 50 (2012), 405-433. doi: 10.1007/s10915-011-9489-z.  Google Scholar

[5]

L. Chen and J. Xu, An optimal streamline diffusion finite element method for a singularly perturbed problem, Recent Advances in Adaptive Computation, Contemp. Math., 383 (2005), 191-201. doi: 10.1090/conm/383/07164.  Google Scholar

[6]

B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), 264-285. doi: 10.1137/S0036142900371544.  Google Scholar

[7]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin finite element method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.  Google Scholar

[8]

P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Robust Computational Techniques For Boundary Layers," Applied Mathematics 16, Chapman & Hall/CRC, Boca Raton, FL, 2000.  Google Scholar

[9]

S. Franz and H-G. Roos, The capriciousness of numerical methods for singular perturbations SIAM Rev., 53 (2011), 157-173. doi: 10.1137/090757344.  Google Scholar

[10]

P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163. doi: 10.1137/S0036142900374111.  Google Scholar

[11]

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), 1-26. doi: 10.2307/2008211.  Google Scholar

[12]

S. Larsson and V. Thomée, "Partial Differential Equations With Numerical Methods," Texts in Applied Mathematics, 45, Springer-Verlag, Berlin, 2003.  Google Scholar

[13]

P. Lesiant and P. A. Raviart, On a finite element method for solving the neutron transport equation, in "Mathematical Aspects of Finite Elements in Partial Differential Equations" (Edited by Carl de Boor), Academic Press, New York-London, (1974).  Google Scholar

[14]

R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions, SIAM J. Numer. Anal., 47 (2008), 89-108. doi: 10.1137/070700267.  Google Scholar

[15]

K. W. Morton, "Numerical Solution of Convection-Diffusion Problems," Applied Mathematics and Mathematical Computation 12, Chapman & Hall, London, 1996.  Google Scholar

[16]

H. G. Roos, M. Stynes and L. Tobiska, "Robust Numerical Methods for Singularly Perturbed Differential Equations," 2nd edition, Springer Series in Computational Mathematics 24, Springer-Verlag, Berlin, 2008.  Google Scholar

[17]

H. G. Roos and H. Zarin, A supercloseness result for the discontinuous Galerkin stabilization of convection-diffusion problems on Shishkin meshes, Num. Meth. Part. Diff. Eq., 23 (2007), 1560-1576. doi: 10.1002/num.20241.  Google Scholar

[18]

D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 38 (2000), 837-875. doi: 10.1137/S0036142999352394.  Google Scholar

[19]

Z. Xie and Z. Zhang, Superconvergence of DG method for one-dimensional singularly perturbed problems, J. Comput. Math., 25 (2007), 185-200.  Google Scholar

[20]

Z. Xie, Z.-Z. Zhang and Z. Zhang, A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems, J. Comp. Math., 27 (2009), 280-298.  Google Scholar

[21]

H. Zarin and H.-G. Roos, Interior penalty discontinuous approximations of convection-diffusion problems with parabolic layers, Numer. Math., 100 (2005), 735-759. doi: 10.1007/s00211-005-0598-1.  Google Scholar

[22]

Z. Zhang, Finite element superconvergence approximation for one-dimensional singularly perturbed problems, Numer. Methods Partial Differential Equations., 18 (2002), 374-395. doi: 10.1002/num.10001.  Google Scholar

[23]

H. Zhu, H. Tian and Z. Zhang, Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems, Comm. Math. Sci., 9 (2011), 1013-1032.  Google Scholar

[24]

H. Zhu and Z. Zhang, Pointwise Error Estimates for the LDG Method Applied to 1-d Singularly Perturbed Reaction-Diffusion Problems,, Comput. Methods Appl. Math., ().  doi: 10.1515/cmam--2012--0004.  Google Scholar

show all references

References:
[1]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.  Google Scholar

[2]

P. Castillo, B. Cockburn, D. Sch$\ddoto$tzau and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Math. Comp., 71 (2002), 455-478. doi: 10.1090/S0025-5718-01-01317-5.  Google Scholar

[3]

F. Celiker and B. Cockburn, Superconvergence of the numerical traces of discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Math. Comp., 76 (2007), 67-96. doi: 10.1090/S0025-5718-06-01895-3.  Google Scholar

[4]

F. Celiker, Z. Zhang and H. Zhu, Nodal superconvergence of SDFEM for singularly perturbed problems, J. Sci. Comput., 50 (2012), 405-433. doi: 10.1007/s10915-011-9489-z.  Google Scholar

[5]

L. Chen and J. Xu, An optimal streamline diffusion finite element method for a singularly perturbed problem, Recent Advances in Adaptive Computation, Contemp. Math., 383 (2005), 191-201. doi: 10.1090/conm/383/07164.  Google Scholar

[6]

B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), 264-285. doi: 10.1137/S0036142900371544.  Google Scholar

[7]

B. Cockburn and C. W. Shu, The local discontinuous Galerkin finite element method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.  Google Scholar

[8]

P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Robust Computational Techniques For Boundary Layers," Applied Mathematics 16, Chapman & Hall/CRC, Boca Raton, FL, 2000.  Google Scholar

[9]

S. Franz and H-G. Roos, The capriciousness of numerical methods for singular perturbations SIAM Rev., 53 (2011), 157-173. doi: 10.1137/090757344.  Google Scholar

[10]

P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), 2133-2163. doi: 10.1137/S0036142900374111.  Google Scholar

[11]

C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), 1-26. doi: 10.2307/2008211.  Google Scholar

[12]

S. Larsson and V. Thomée, "Partial Differential Equations With Numerical Methods," Texts in Applied Mathematics, 45, Springer-Verlag, Berlin, 2003.  Google Scholar

[13]

P. Lesiant and P. A. Raviart, On a finite element method for solving the neutron transport equation, in "Mathematical Aspects of Finite Elements in Partial Differential Equations" (Edited by Carl de Boor), Academic Press, New York-London, (1974).  Google Scholar

[14]

R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions, SIAM J. Numer. Anal., 47 (2008), 89-108. doi: 10.1137/070700267.  Google Scholar

[15]

K. W. Morton, "Numerical Solution of Convection-Diffusion Problems," Applied Mathematics and Mathematical Computation 12, Chapman & Hall, London, 1996.  Google Scholar

[16]

H. G. Roos, M. Stynes and L. Tobiska, "Robust Numerical Methods for Singularly Perturbed Differential Equations," 2nd edition, Springer Series in Computational Mathematics 24, Springer-Verlag, Berlin, 2008.  Google Scholar

[17]

H. G. Roos and H. Zarin, A supercloseness result for the discontinuous Galerkin stabilization of convection-diffusion problems on Shishkin meshes, Num. Meth. Part. Diff. Eq., 23 (2007), 1560-1576. doi: 10.1002/num.20241.  Google Scholar

[18]

D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal., 38 (2000), 837-875. doi: 10.1137/S0036142999352394.  Google Scholar

[19]

Z. Xie and Z. Zhang, Superconvergence of DG method for one-dimensional singularly perturbed problems, J. Comput. Math., 25 (2007), 185-200.  Google Scholar

[20]

Z. Xie, Z.-Z. Zhang and Z. Zhang, A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems, J. Comp. Math., 27 (2009), 280-298.  Google Scholar

[21]

H. Zarin and H.-G. Roos, Interior penalty discontinuous approximations of convection-diffusion problems with parabolic layers, Numer. Math., 100 (2005), 735-759. doi: 10.1007/s00211-005-0598-1.  Google Scholar

[22]

Z. Zhang, Finite element superconvergence approximation for one-dimensional singularly perturbed problems, Numer. Methods Partial Differential Equations., 18 (2002), 374-395. doi: 10.1002/num.10001.  Google Scholar

[23]

H. Zhu, H. Tian and Z. Zhang, Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems, Comm. Math. Sci., 9 (2011), 1013-1032.  Google Scholar

[24]

H. Zhu and Z. Zhang, Pointwise Error Estimates for the LDG Method Applied to 1-d Singularly Perturbed Reaction-Diffusion Problems,, Comput. Methods Appl. Math., ().  doi: 10.1515/cmam--2012--0004.  Google Scholar

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