Convergence Analysis of a Discontinuous Galerkin Method on Bakhvalov-Type Meshes for a First-Order Nonlinear Singularly Perturbed Delay Problem

Abstract

In this paper, a discontinuous Galerkin (DG) method is proposed to solve a first-order nonlinear singularly perturbed delay differential equation (SPDDE). At first, the construction of a generalized Bakhvalov-type (B-type) mesh and the corresponding characteristics are given. Furthermore, an optimal parameter-uniform convergence rate

s+1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s+1$$\end{document}

in the

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

-norm is proved, where s is the degree of the piecewise polynomial space. Finally, two numerical experiments are carried out to complement the theoretical results of our proposed numerical method.

Keywords

Singularly perturbed problems (SPPs) / Discontinuous Galerkin (DG) / Bakhvalov mesh / Error analysis / 65L20 / 65L50 / 65L70

References

[1] Amiraliyev GM, Erdogan F. Uniform numerical method for singularly perturbed delay differential equations. Comput. Math. Appl., 2007, 53: 1251-1259

[2] Amiraliyev GM, Erdogan F. A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations. Numer. Algorithms, 2009, 52: 663-675

[3] Amiraliyeva IG, Amiraliyev GM. Uniform difference method for parameterized singularly perturbed delay differential equations. Numer. Algorithms, 2009, 52: 509-521

[4] Bakhvalov NS. The optimization of methods of solving boundary value problems with a boundary layer. USSR Comput. Math. Math. Phys., 1969, 9: 139-166

[5] Bawa RK, Lal A, Kumar V. An ε\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varepsilon $$\end{document}-uniform hybrid scheme for singularly perturbed delay differential equations. Appl. Math. Comput., 2011, 217: 8216-8222

[6] Cen Z. A second-order finite difference scheme for a class of singularly perturbed delay differential equations. Int. J. Comput. Math., 2010, 87: 173-185

[7] Derstine MW, Gibbs HM, Hopf FA, Kaplan DL. Bifurcation gap in a hybrid optically bistable system. Phys. Rev. A, 1982, 26: 3720-3722

[8] Erdogan F, Cen Z. A uniformly almost second order convergent numerical method for singularly perturbed delay differential equations. J. Comput. Appl. Math., 2018, 333: 382-394

[9] Helena Z. On discontinuous Galerkin finite element method for singularly perturbed delay differential equations. Appl. Math. Lett., 2014, 38: 27-32

[10] Howlader J, Mishra P, Sharma KK. An orthogonal spline collocation method for singularly perturbed parabolic reaction-diffusion problems with time delay. J. Appl. Math. Comput., 2024, 70: 1069-1101

[11] Howlader J, Mishra P, Sharma KK. A parameter uniform orthogonal spline collocation method for time delay singularly perturbed semilinear reaction-diffusion problems. J. Appl. Math. Comput., 2025, 71: 1077-1107

[12] Kabeto MJ, Duressa GF. Robust numerical method for singularly perturbed semilinear parabolic differential difference equations. Math. Comput. Simul., 2021, 188: 537-547

[13] Kumar S. Layer-adapted methods for quasilinear singularly perturbed delay differential problems. Appl. Math. Comput., 2014, 233: 214-221

[14] Kumar S, Kumar M. A second order uniformly convergent numerical scheme for parameterized singularly perturbed delay differential problems. Numer. Algorithms, 2017, 76: 349-360

[15] Linß TLayer-Adapted Meshes for Reaction-Convection-Diffusion Problems, 1998BerlinSpringer

[16] Liu Y, Cheng Y. Local discontinuous Galerkin method for a singularly perturbed fourth-order problem of reaction-diffusion type. J. Comput. Appl. Math., 2024, 440 Article ID: 115641

[17] Longtin A, Milton JG. Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Math. Biosci., 1988, 90: 183-199

[18] Mackey M, Glass L. Oscillation and chaos in physiological control systems. Science, 1977, 197: 287-289

[19] Mallet-Paret J, Nussbaum R. A differential-delay equation arising in optics and physiology. SIAM J. Math. Anal., 1989, 20: 249-292

[20] Nhan TA, Vulanović R. Analysis of a second-order hybrid scheme on Bakhvalov-type meshes: the truncation-error and barrier-function approach. Appl. Numer. Math., 2023, 186: 84-99

[21] Rajeev Ranjan K, Gowrisankar S. Uniformly convergent NIPG method for singularly perturbed convection diffusion problem on Shishkin type meshes. Appl. Numer. Math., 2022, 179: 125-148

[22] Ranjan R. A novel fitted spline method for the numerical treatment of singularly perturbed differential equations having small delays. J. Appl. Math. Comput., 2023, 69: 4645-4664

[23] Roos H-G, Stynes M, Tobiska LNumerical Methods for Singularly Perturbed Differential Equations, 2008BerlinSpringer

[24] Tao X, Xie Z. The uniform convergence of a DG method for a singularly perturbed Volterra integro-differential equation. Acta. Math. Sci., 2023, 43: 2159-2178

[25] Vulanović R. On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh.. Univ. u Novom Sadu Zb Rad. Prir. Mat. Fak. Ser. Mat., 1983, 13: 187-201

[26] Xie Z, Zhang Z. Superconvergence of DG method for one-dimensional singularly perturbed problems. J. Comput. Math., 2007, 25: 185-200

[27] Xie Z, Zhang Z. Uniform superconvergence analysis of the discontinuous Galerkin method for a singularly perturbed problem in 1-D. Math. Comput., 2010, 79: 35-45

[28] Zhang C, Zhang J, Zheng W. Supercloseness of the LDG method for a singularly perturbed convection-diffusion problem on Bakhvalov-type mesh. Appl. Math. Lett., 2024, 151 Article ID: 108989

[29] Zhang J, Liu X. Uniformconvergence of a weak Galerkin method for singularly perturbed convection-diffusion problems. Math. Comput. Simul., 2022, 200: 393-403

[30] Zhu H, Celiker F. Nodal superconvergence of the local discontinuous Galerkin method for singularly perturbed problems. J. Comput. Appl. Math., 2018, 330: 95-116

[31] Zhu P, Xie S. Higher order uniformly convergent continuous/discontinuous Galerkin methods for singularly perturbed problems of convection-diffusion type. Appl. Numer. Math., 2014, 76: 48-59

[32] Zhu P, Xie Z, Zhou S. A uniformly convergent continuous-discontinuous Galerkin method for singularly perturbed problems of convection-diffusion type. Appl. Math. Comput., 2011, 217: 4781-4790

Funding

National Natural Science Foundation of China(12361087)

RIGHTS & PERMISSIONS

Shanghai University