American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.  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, 383 (2005), 191.  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.  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.  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, (2000).   Google Scholar

[9]

S. Franz and H-G. Roos, The capriciousness of numerical methods for singular perturbations, SIAM Rev., 53 (2011), 157.  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.  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.  doi: 10.2307/2008211.  Google Scholar

[12]

S. Larsson and V. Thomée, "Partial Differential Equations With Numerical Methods,", Texts in Applied Mathematics, 45 (2003).   Google Scholar

[13]

P. Lesiant and P. A. Raviart, On a finite element method for solving the neutron transport equation,, in, (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.  doi: 10.1137/070700267.  Google Scholar

[15]

K. W. Morton, "Numerical Solution of Convection-Diffusion Problems,", Applied Mathematics and Mathematical Computation 12, (1996).   Google Scholar

[16]

H. G. Roos, M. Stynes and L. Tobiska, "Robust Numerical Methods for Singularly Perturbed Differential Equations,", 2nd edition, (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.  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.  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.   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.   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.  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.  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.   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.  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.  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.  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.  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, 383 (2005), 191.  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.  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.  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, (2000).   Google Scholar

[9]

S. Franz and H-G. Roos, The capriciousness of numerical methods for singular perturbations, SIAM Rev., 53 (2011), 157.  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.  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.  doi: 10.2307/2008211.  Google Scholar

[12]

S. Larsson and V. Thomée, "Partial Differential Equations With Numerical Methods,", Texts in Applied Mathematics, 45 (2003).   Google Scholar

[13]

P. Lesiant and P. A. Raviart, On a finite element method for solving the neutron transport equation,, in, (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.  doi: 10.1137/070700267.  Google Scholar

[15]

K. W. Morton, "Numerical Solution of Convection-Diffusion Problems,", Applied Mathematics and Mathematical Computation 12, (1996).   Google Scholar

[16]

H. G. Roos, M. Stynes and L. Tobiska, "Robust Numerical Methods for Singularly Perturbed Differential Equations,", 2nd edition, (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.  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.  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.   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.   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.  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.  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.   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

[1]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817
[2]

Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104
[3]

Rongjie Lai, Jiang Liang, Hong-Kai Zhao. A local mesh method for solving PDEs on point clouds. Inverse Problems & Imaging, 2013, 7 (3) : 737-755. doi: 10.3934/ipi.2013.7.737
[4]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
[5]

Vyacheslav K. Isaev, Vyacheslav V. Zolotukhin. Introduction to the theory of splines with an optimal mesh. Linear Chebyshev splines and applications. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 471-489. doi: 10.3934/naco.2013.3.471
[6]

Yulong Xing, Ching-Shan Chou, Chi-Wang Shu. Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Problems & Imaging, 2013, 7 (3) : 967-986. doi: 10.3934/ipi.2013.7.967
[7]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[8]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139
[9]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 0 (0) : 0-0. doi: 10.3934/era.2020078
[10]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241
[11]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719
[12]

Qingjie Hu, Zhihao Ge, Yinnian He. Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2923-2948. doi: 10.3934/dcdsb.2020046
[13]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
[14]

Xia Ji, Wei Cai. Accurate simulations of 2-D phase shift masks with a generalized discontinuous Galerkin (GDG) method. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 401-415. doi: 10.3934/dcdsb.2011.15.401
[15]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
[16]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[17]

Houssem Haddar, Alexander Konschin. Factorization method for imaging a local perturbation in inhomogeneous periodic layers from far field measurements. Inverse Problems & Imaging, 2020, 14 (1) : 133-152. doi: 10.3934/ipi.2019067
[18]

Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022
[19]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[20]

Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic & Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences