Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems

Year 2023, Volume: 52 Issue: 4, 850 - 875, 15.08.2023

Şuayip Toprakseven

https://doi.org/10.15672/hujms.1117320

Cited By: 4

Abstract

In this paper, we propose a weak Galerkin finite element method (WG-FEM) for solving two-point boundary value problems of convection-dominated type on a Bakhvalov-type mesh. A special interpolation operator which has a simple representation and can be easily extended to higher dimensions is introduced for convection-dominated problems. A robust optimal order of uniform convergence has been proved in the energy norm with this special interpolation using piecewise polynomials of degree $k\geq 1$ on interior of the elements and piecewise constant on the boundary of each element. The proposed finite element scheme is {parameter-free formulation} and since the interior degrees of freedom can be eliminated efficiently from the resulting discrete system, the number of unknowns of the proposed method is comparable with the standard finite element methods. An optimal order of uniform convergence is derived on Bakhvalov-type mesh. Finally, numerical experiments are given to support the theoretical findings and to show the efficiency of the proposed method.

Keywords

singularly perturbed differential equation, weak Galerkin finite element method, Bakhvalov-type meshes, optimal uniform convergence

References

• [1] V.B. Andreev and N. Kopteva, On the convergence, uniform with respect to a small parameter, of monotone three-point difference schemes, Differ. Equ. 34 (7), (1998).
• [2] N.S. Bakhvalov, On the optimization of the methods for solving boundary value problems in the presence of a boundary layer, Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859, 1969.
• [3] M. Bradar and H. Zarin, A singularly perturbed problem with two parameters on a Bakhvalov-type mesh, J. Comput. Appl. Math. 292, 307–319, 2016.
• [4] P.G. Ciarlet, The finite element method for elliptic problems, SIAM, 2002.
• [5] S. Franz and H.-G Roos, The capriciousness of numerical methods for singular perturbations, SIAM Rev. 53 (1), 157–173, 2011.
• [6] N. Kopteva, On the convergence, uniform with respect to the small parameter, of a scheme with central difference on refined grids, Zh. Vychisl. Mat. Mat. Fiz. 39 (10), 1662–1678, 1999.
• [7] N. Kopteva, Uniform pointwise convergence of difference schemes for convectiondiffusion problems on layer-adapted meshes, Computing, 66, 179–197, 2001.
• [8] R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reactiondiffusion problems in one and two dimensions, SIAM J. Numer. Anal. 47, 89–108, 2008.
• [9] R. Lin, Discontinuous Galerkin least-squares finite element methods for singularly perturbed reactiondiffusion problems with discontinuous coefficients and boundary singularities, Numer. Math. 112, 295–318, 2009.
• [10] R. Lin, X. Ye, S. Zhang and P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusionreaction problems, SIAM J. Numer. Anal. 56 (3), 1482–1497, 2018.
• [11] T. Linss, Layeradapted meshes for reactionconvectiondiffusion problems, In Lecture Notes in Mathematics, vol. 1985. Springer, Berlin 2010.
• [12] T. Linss, The necessity of Shishkin decompositions, Appl. Math. Lett. 14, 891–896, 2001.
• [13] T. Linss and M. Stynes, The SDFEM on Shishkin meshes for linear convectiondiffusion problems, Numer. Math. 87, 457–484, 2001.
• [14] T. Linss and H.-G. Ross, Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters, J. Math. Anal. Appl. 289, 355–366, 2004.
• [15] L. Liu, H. Leng and G. Long, Analysis of the SDFEM for singularly perturbed differentialdifference equations, Calcolo 55 (3), 1–17, 2018.
• [16] J. M. Melenk,hp-Finite Element Methods for Singular Perturbations, In: Lecture Notes in Mathematics, vol. 1796. Springer, Berlin 2002.
• [17] J.J.H. Miller, E. ORiordan and G.L. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore 1996.
• [18] L. Mu, J. Wang, X. Ye and S. Zhao, A new weak Galerkin finite element method for elliptic interface problems, J. Comput. Phys. 325, 157–173, 2016.
• [19] L. Mu, J. Wang, X. Ye and S. Zhao,A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput. 65, 363–386, 2015.
• [20] D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, Springer-Verlag, Berlin 2012.
• [21] H.G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion-Reaction and Flow Problems (second edition). In: Springer Series in Computational Mathematics, vol. 24. Springer, Berlin 2008.
• [22] H.-G. Roos and H. Zarin, A supercloseness result for the discontinuous Galerkin stabilization of convectiondiffusion problems on Shishkin meshes, Numer. Methods Partial Differ. Equ. 23 (6), 1560–1576, 2007.
• [23] H. -G. Roos, Error estimates for linear finite elements on Bakhvalov-type meshes, Appl. Math. 51, 63–72, 2006.
• [24] H. -G. Ross and M. Stynes, Some open questions in the numerical analysis of singularly perturbed differential equations, Comput. Methods Appl. Math. 15 (4), 531– 550, 2015.
• [25] M. Stynes and E. ORiorddan, A uniformly convergent Galerkin method on a Shishkin mesh for a convectiondiffusion problem, J. Math. Anal. Appl. 214, 36–54, 1997.
• [26] M. Stynes and L. Tobiska, Analysis of the streamline-diffusion finite element method on a Shishkin mesh for a convectiondiffusion problem with exponential layers, J. Numer. Math. 9, 59–76, 2001.
• [27] S. Sumit, S. Kumar and M. Kumar, Optimal fourth-order parameter-uniform convergence of a non-monotone scheme on equidistributed meshes for singularly perturbed reactiondiffusion problems, Int. J. Comput. Math. 1–16, 2021.
• [28] S. Toprakseven, A weak Galerkin finite element method for time fractional reactiondiffusion- convection problems with variable coefficients, Appl. Numer. Math. 168, 1–12, 2021.
• [29] S. Toprakseven and P. Zhu, Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems, Turkish J. Math. 45 (6), 2703–2730, 2021.
• [30] S. Toprakseven, Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems, Calcolo 59 (1), 1–35, 2022.
• [31] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math. 241, 103–115, 2013.
• [32] J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math. 42, 155–174, 2016.
• [33] H. Zarin and H.-G. Roos, Interior penalty discontinuous approximations of convectiondiffusion problems with parabolic layers, Numer. Math. 100, 735–759, 2005.
• [34] Z. Zhang, Finite element superconvergence on Shishkin mesh for 2D convectiondiffusion problems, Math. Comput. 245, 1147–1177, 2003.
• [35] J. Zhang and X. Liu, Optimal order of uniform convergence for finite element method on Bakhvalov-type meshes, J. Sci. Comput. 85 (1), 1–14, 2020.
• [36] J. Zhang and Y. Lv, High-order finite element method on a Bakhvalov-type mesh for a singularly perturbed convectiondiffusion problem with two parameters, Appl. Math. Comput. 397, 125953, 2021.
• [37] P. Zhu, Y. Yang and Y. Yin, Higher order uniformly convergent NIPG methods for 1-d singularly perturbed problems of convection-diffusion type, Appl. Math. Model. 39 (22), 6806–6816, 2015.
• [38] H. Zhu and Z. Zhang, Uniform convergence of the LDG method for a singularly perturbed problem with the exponential boundary layer, Math. Comput. 83, 635–663, 2014.
• [39] P. Zhu and S. Xie, A uniformly convergent weak Galerkin finite element method on Shishkin mesh for 1d convectiondiffusion problem, J. Sci. Comput. 85 (2), 1–22, 2020.