Home Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems
Article
Licensed
Unlicensed Requires Authentication

Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems

  • Yao Cheng ORCID logo EMAIL logo , Chuanjing Song and Yanjie Mei
Published/Copyright: September 1, 2020

Abstract

Local discontinuous Galerkin method is considered for time-dependent singularly perturbed semilinear problems with boundary layer. The method is equipped with a general numerical flux including two kinds of independent parameters. By virtue of the weighted estimates and suitably designed global projections, we establish optimal (k+1) -th error estimate in a local region at a distance of 𝒪(hlog(1h)) from domain boundary. Here k is the degree of piecewise polynomials in the discontinuous finite element space and h is the maximum mesh size. Both semi-discrete LDG method and fully discrete LDG method with a third-order explicit Runge–Kutta time-marching are considered. Numerical experiments support our theoretical results.

MSC 2010: 65M12; 65M15; 65M60

Award Identifier / Grant number: 11801396

Award Identifier / Grant number: 11802193

Award Identifier / Grant number: BK20170374

Funding statement: The first and third author were funded by National Natural Science Foundation of China grant 11801396, National Science Foundation of Jiangsu Province grant BK20170374 and Natural Science Foundation of the Jiangsu Higher Education Institution of China grant 17KJB110016. The second author was funded by National Natural Science Foundation of China grant 11802193 and Natural Science Foundation of the Jiangsu Higher Education Institution of China grant 18KJB130005.

Acknowledgements

The authors gratefully acknowledge the editor and two anonymous referees for their time and invaluable suggestions, which improve the quality of the paper to a large extent.

References

[1] J. L. Bona, H. Chen, O. Karakashian and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg–de Vries equation, Math. Comp. 82 (2013), no. 283, 1401–1432. 10.1090/S0025-5718-2013-02661-0Search in Google Scholar

[2] P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 38 (2000), no. 5, 1676–1706. 10.1137/S0036142900371003Search in Google Scholar

[3] P. Castillo, B. Cockburn, D. Schötzau 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), no. 238, 455–478. 10.1090/S0025-5718-01-01317-5Search in Google Scholar

[4] Y. Cheng, Optimal error estimate of the local discontinuous Galerkin methods based on the generalized alternating numerical fluxes for nonlinear convection-diffusion equations, Numer. Algorithms 80 (2019), no. 4, 1329–1359. 10.1007/s11075-018-0529-8Search in Google Scholar

[5] Y. Cheng, X. Meng and Q. Zhang, Application of generalized Gauss–Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp. 86 (2017), no. 305, 1233–1267. 10.1090/mcom/3141Search in Google Scholar

[6] Y. Cheng and Q. Zhang, Local analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem, J. Sci. Comput. 72 (2017), no. 2, 792–819. 10.1007/s10915-017-0378-ySearch in Google Scholar

[7] Y. Cheng, Q. Zhang and H. Wang, Local analysis of the local discontinuous Galerkin method with the generalized alternating numerical flux for two-dimensional singularly perturbed problem, Int. J. Numer. Anal. Model. 15 (2018), no. 6, 785–810. Search in Google Scholar

[8] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. 10.1137/S0036142997316712Search in Google Scholar

[9] S. Gowrisankar and S. Natesan, The parameter uniform numerical method for singularly perturbed parabolic reaction-diffusion problems on equidistributed grids, Appl. Math. Lett. 26 (2013), no. 11, 1053–1060. 10.1016/j.aml.2013.05.017Search in Google Scholar

[10] J. Guzmán, Local analysis of discontinuous Galerkin methods applied to singularly perturbed problems, J. Numer. Math. 14 (2006), no. 1, 41–56. 10.1515/156939506776382157Search in Google Scholar

[11] N. Heuer and M. Karkulik, A robust DPG method for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal. 55 (2017), no. 3, 1218–1242. 10.1137/15M1041304Search in Google Scholar

[12] C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1–3, 285–312. 10.1016/0045-7825(84)90158-0Search in Google Scholar

[13] N. Kopteva, Maximum norm a posteriori error estimate for a 2D singularly perturbed semilinear reaction-diffusion problem, SIAM J. Numer. Anal. 46 (2008), no. 3, 1602–1618. 10.1137/060677616Search in Google Scholar

[14] J. Li, Convergence analysis of finite element methods for singularly perturbed problems, Comput. Math. Appl. 40 (2000), no. 6–7, 735–745. 10.1016/S0898-1221(00)00192-9Search in Google Scholar

[15] T. Linß and N. Madden, Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems, IMA J. Numer. Anal. 29 (2009), no. 1, 109–125. 10.1093/imanum/drm053Search in Google Scholar

[16] H. Liu and N. Ploymaklam, A local discontinuous Galerkin method for the Burgers–Poisson equation, Numer. Math. 129 (2015), no. 2, 321–351. 10.1007/s00211-014-0641-1Search in Google Scholar

[17] X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations, Math. Comp. 85 (2016), no. 299, 1225–1261. 10.1090/mcom/3022Search in Google Scholar

[18] J. J. H. Miller, E. O’Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, River Edge, 1996. 10.1142/2933Search in Google Scholar

[19] W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, 1973. Search in Google Scholar

[20] H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 2008. Search in Google Scholar

[21] A. H. Schatz and L. B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp. 40 (1983), no. 161, 47–89. 10.1090/S0025-5718-1983-0679434-4Search in Google Scholar

[22] Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys. 7 (2010), no. 1, 1–46. Search in Google Scholar

[23] H. Zhu and Z. Zhang, Local error estimates of the LDG method for 1-D singularly perturbed problems, Int. J. Numer. Anal. Model. 10 (2013), no. 2, 350–373. Search in 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. 13 (2013), no. 1, 79–94. 10.1515/cmam-2012-0004Search in Google Scholar

Received: 2019-12-14
Revised: 2020-05-05
Accepted: 2020-08-05
Published Online: 2020-09-01
Published in Print: 2021-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 26.4.2025 from https://www.degruyterbrill.com/document/doi/10.1515/cmam-2019-0185/html
Scroll to top button