# American Institute of Mathematical Sciences

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

 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.
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] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955 [18] Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783 [19] Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034 [20] Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051

2018 Impact Factor: 1.008