Reliable and Efficient a Posteriori Error Analysis of the Weak Galerkin FEM for Singularly Perturbed Two-Dimensional Reaction-Diffusion Problems

Abstract

In this research work, we investigate a residual-based approach for a posteriori error estimation within the framework of the weak Galerkin finite element method (WGFEM), focusing on singularly perturbed reaction-diffusion problems in the energy norm. Capturing the layer behaviour in the reaction-dominated regime is more complicated compared to the convection-dominated regime, a weighted robust estimator is employed for accurately capturing the layer behaviour for this model problem, which has a reaction-dominated regime. The global reliability and efficiency are verified by deriving both the upper and lower bounds of the proposed estimator. An adaptive WGFEM is developed using this estimator, employing a mesh refinement strategy to effectively resolve the layers for the complicated reaction-dominated regime. Theoretically, the proposed estimator is both reliable and efficient, while numerical results on the unit square and the unit circular domains confirm its effectiveness.

Keywords

A posteriori error estimation / Singularly perturbed problems (SPPs) / Reaction-diffusion equations / Boundary layers / Weak Galerkin finite element method (WGFEM) / Adaptive mesh refinement / 65N50 / 35J25 / 35B25 / 65N30

References

[1] Ainsworth, M., Babuska, I.: Reliable and robust a posteriori error estimation for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 36, 331–353 (1999)

[2] Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. Teubner, Stuttgart (1999)

[3] Cai, D., Cai, Z.: Hybrid a posteriori error estimators for conforming finite element approximations to stationary convection-diffusion-reaction equations. Numer. Math. 157, 477–504 (2025)

[4] Chen L, Wang J, Ye X. A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems. J. Sci. Comput., 2014, 59: 496-511

[5] Clément P. Approximation by finite element functions using local regularization, Revue française d’automatique, informatique, recherche opérationnelle. Anal. Numérique, 1975, 9: 77-84

[6] Dörfler W. A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal., 1996, 33: 1106-1124

[7] Funken, S., Praetorius, D., Wissgott, P.: Efficient implementation of adaptive P1-FEM in MATLAB. ASC Report 19/2008. Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna (2008)

[8] Gaucel, S., Langlais, M.: Some remarks on a singular reaction-diffusion system arising in predator-prey modeling. Discrete Contin. Dyn. Syst. Ser. B 8, 61 (2007)

[9] Hegarty, A.F., O’Riordan, E.: Parameter-uniform numerical method for singularly perturbed convection-diffusion problem on a circular domain. Adv. Comput. Math. 43, 885–909 (2017)

[10] Hong, Y., Jung, C.-Y., Laminie, J.: Singularly perturbed reaction-diffusion equations in a circle with numerical applications. Int. J. Comput. Math. 90, 2308–2325 (2013)

[11] Kopteva, N., Rankin, R.: Pointwise a posteriori error estimates for discontinuous Galerkin methods for singularly perturbed reaction-diffusion equations. SIAM J. Numer. Anal. 61, 1938–1961 (2023)

[12] Ku, J., Stynes, M.: A posteriori error estimates for a dual finite element method for singularly perturbed reaction-diffusion problems. BIT Numer. Math. 64, 7 (2024)

[13] Kumar, S., Natesan, S.: An efficient weak Galerkin finite element method for generalized Black-Scholes PDEs modelling option pricing. Int. J. Comput. Math. 102, 761–778 (2025)

[14] Larson, M.G., Niklasson, A.J.: Conservation Properties for the Continuous and Discontinuous Galerkin Methods, vol. 8. Chalmers Finite Element Center Preprint (2000)

[15] Lin, R., Stynes, M.: A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 50, 2729–2743 (2012)

[16] Linß, T.: Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems. Springer, Berlin (2009)

[17] Pal, A., Natesan, S.: Robust error estimates for weak Galerkin finite element method for singularly perturbed 2D reaction-diffusion elliptic boundary-value problems on various layer-adapted meshes. Numer. Algorithms (2026). https://doi.org/10.1007/s11075-025-02306-3

[18] Pal A, Natesan S. Uniform convergence analysis of dimensional-splitting weak Galerkin FEM and classical weak Galerkin FEM for singularly perturbed 2D nonlinear parabolic PDEs. Commun. Nonlinear Sci. Numer. Simul., 2026, 158 109834

[19] Pao C. Singular reaction diffusion equations of porous medium type. Nonlinear Anal. Theory Methods Appl., 2009, 71: 2033-2052

[20] Raina A, Natesan S. A weak Galerkin finite element method for fourth-order parabolic singularly perturbed problems on layer adapted Shishkin mesh. Appl. Numer. Math., 2025, 207: 520-533

[21] Sun S, Huang Z, Wang C. Weak Galerkin finite element method for a class of quasilinear elliptic problems. Appl. Math. Lett., 2018, 79: 67-72

[22] Toprakseven, S., Zhu, P.: Supercloseness of weak Galerkin methods in a weighted and balanced norm for singularly perturbed reaction-diffusion problems. Math. Comput. Simul. 246, 491–508 (2026)

[23] Verfürth, R.: A posteriori error estimators for convection-diffusion equations. Numer. Math. 80, 641–663 (1998)

[24] Verfürth, R.: Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math. 78, 479–493 (1998)

[25] Wang J, Ye X. A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math., 2013, 241: 103-115

[26] Zhang J, Liu X. Uniform convergence of a weak Galerkin finite element method on Shishkin mesh for singularly perturbed convection-diffusion problems in 2D. Appl. Math. Comput., 2022, 432127346

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