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

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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

Abstract
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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 and 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.
[2]	P. Castillo, B. Cockburn, D. Sch$\ddot{O}$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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	S. Larsson and V. Thomée, "Partial Differential Equations With Numerical Methods," Texts in Applied Mathematics, 45, Springer-Verlag, Berlin, 2003.
[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).
[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.
[15]	K. W. Morton, "Numerical Solution of Convection-Diffusion Problems," Applied Mathematics and Mathematical Computation 12, Chapman & Hall, London, 1996.
[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.
[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.
[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.
[19]	Z. Xie and Z. Zhang, Superconvergence of DG method for one-dimensional singularly perturbed problems, J. Comput. Math., 25 (2007), 185-200.
[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.
[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.
[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.
[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.
[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., Published Online: 10/19/2012. doi: 10.1515/cmam--2012--0004.

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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.
[2]	P. Castillo, B. Cockburn, D. Sch$\ddot{O}$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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	S. Larsson and V. Thomée, "Partial Differential Equations With Numerical Methods," Texts in Applied Mathematics, 45, Springer-Verlag, Berlin, 2003.
[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).
[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.
[15]	K. W. Morton, "Numerical Solution of Convection-Diffusion Problems," Applied Mathematics and Mathematical Computation 12, Chapman & Hall, London, 1996.
[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.
[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.
[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.
[19]	Z. Xie and Z. Zhang, Superconvergence of DG method for one-dimensional singularly perturbed problems, J. Comput. Math., 25 (2007), 185-200.
[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.
[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.
[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.
[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.
[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., Published Online: 10/19/2012. doi: 10.1515/cmam--2012--0004.

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