Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

Şuayip Toprakseven; Seza Dinibutun; Şuayip Toprakseven; Seza Dinibutun

doi:10.3934/math.2023788

AIMS Mathematics

2023, Volume 8, Issue 7: 15427-15465. doi: 10.3934/math.2023788

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Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

Şuayip Toprakseven ^{1
,
,},
Seza Dinibutun ²

1.
Artvin Vocational School, Accounting and Taxation, Artvin Çoruh University, Artvin, 08100, Turkey
2.
International University of Kuwait, Ardiya, Kuwait Department of Mathematics and Natural Sciences, Kuwait

Received: 13 March 2023 Revised: 10 April 2023 Accepted: 13 April 2023 Published: 26 April 2023
MSC : 35J50, 65N15, 65N30

This paper introduces a weak Galerkin ﬁnite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.
Citation: Şuayip Toprakseven, Seza Dinibutun. Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms[J]. AIMS Mathematics, 2023, 8(7): 15427-15465. doi: 10.3934/math.2023788

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Abstract

This paper introduces a weak Galerkin ﬁnite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.

References

[1]	H. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Berlin, Heidelberg: Springer, 2008. https://doi.org/10.1007/978-3-540-34467-4
[2]	T. Linss, Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Berlin, Heidelberg: Springer, 2008, https://doi.org/10.1007/978-3-642-05134-0
[3]	N. S. Bakhvalov, The optimization of methods of solving boundary value problems with a boundary layer, USSR Comput. Math. Math. Phys., 9 (1969), 139–166. https://doi.org/10.1016/0041-5553(69)90038-X doi: 10.1016/0041-5553(69)90038-X
[4]	J. Miller, E. O'Riordan, G. Shishkin, Fitted Numerical Methods For Singular Perturbation Problems, Singapore: World Scientific, 1996. https://doi.org/10.1142/8410
[5]	G. Shishkin, Mesh approximation of singularly perturbed boundary-value problems for systems of elliptic and parabolic equations, Comput. Math. Math. Phys., 35 (1995), 429–446.
[6]	N. Madden, M. Stynes, A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems, IMA J. Numer. Anal., 23 (2003), 627– 644. https://doi.org/10.1093/imanum/23.4.627 doi: 10.1093/imanum/23.4.627
[7]	N. Madden, M. Stynes, A finite element analysis of a coupled system of singularly perturbed reaction-diffusion equations, Appl. Math. Comput., 148 (2004), 869–880. https://doi.org/10.1016/S0096-3003(02)00955-4 doi: 10.1016/S0096-3003(02)00955-4
[8]	T. Linss, N. Madden, Accurate solution of a system of coupled singularly perturbed reaction- diffusion equations, Computing, 73 (2004), 121–133. https://doi.org/10.1007/s00607-004-0065-3 doi: 10.1007/s00607-004-0065-3
[9]	T. Linss, N. Madden, Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems, IMA J. Numer. Anal., 29 (2009), 109–125. https://doi.org/10.1093/imanum/drm053 doi: 10.1093/imanum/drm053
[10]	T. Linss, Layer-adapted meshes for one-dimensional reaction-convection-diffusion problems, J. Numer. Math., 12 (2004), 193–205. https://doi.org/10.1515/1569395041931482 doi: 10.1515/1569395041931482
[11]	L. Tobiska, Analysis of a new stabilized higher order finite element method for advection-diffusion equations, Comput. Methods Appl. Mech. Eng., 196 (2006), 538–550. https://doi.org/10.1016/j.cma.2006.05.009 doi: 10.1016/j.cma.2006.05.009
[12]	S. Natesan, B. Deb, A robust computational method for singularly perturbed coupled system of reaction-diffusion boundary-value problems, Appl. Math. Comput., 188 (2007), 353–364. https://doi.org/10.1016/j.amc.2006.09.120 doi: 10.1016/j.amc.2006.09.120
[13]	P. Das, S. Natesan, Error estimate using mesh equidistribution technique for singularly perturbed system of reaction-diffusion boundary-value problems, Appl. Math. Comput., 249 (2014), 265–277. https://doi.org/10.1016/j.amc.2014.10.023 doi: 10.1016/j.amc.2014.10.023
[14]	R. Lin, M. Stynes, A balanced finite element method for a system of singularly perturbed reaction-diffusion two-point boundary value problems, Numer. Algor., 70 (2015), 691–707. https://doi.org/10.1007/s11075-015-9969-6 doi: 10.1007/s11075-015-9969-6
[15]	R. B. Kellogg, N. Madden, M. Stynes, A parameter-robust numerical method for a system of reaction-diffusion equations in two dimensions, Numer. Methods Partial Differ. Equ., 24, 335–348. https://doi.org/10.1002/num.20265
[16]	S. Franz, H. G. Roos, Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers, Calcolo, 51 (2016), 105–132. https://doi.org/10.1007/s10092-013-0093-5 doi: 10.1007/s10092-013-0093-5
[17]	J. M. Melenk, C. Xenophontos, Robust exponential convergence of hp-Fem in balanced norms for singularly perturbed reaction-diffusion equations, Calcolo, 53 (2014), 423–440. https://doi.org/10.1007/s10092-015-0139-y doi: 10.1007/s10092-015-0139-y
[18]	H. G. Roos, M. Schopf, Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems, ZAMM, 95 (2015), 551–565. https://doi.org/10.1002/zamm.201300226 doi: 10.1002/zamm.201300226
[19]	N. Madden, M. Stynes, A weighted and balanced fem for singularly perturbed reaction-diffusion problems, Calcolo, 58 (2021), 28. https://doi.org/10.13140/RG.2.2.26317.87525 doi: 10.13140/RG.2.2.26317.87525
[20]	J. Zhang, X. Liu, Convergence and supercloseness in a balanced norm of finite element methods on Bakhvalov-type meshes for reaction-diffusion problems, J. Sci. Comput., 88 (2021), 27. https://doi.org/10.1007/s10915-021-01542-8 doi: 10.1007/s10915-021-01542-8
[21]	H. G. Roos, Error estimates in balanced norms of finite element methods on layer-adapted meshes for second order reaction-diffusion problems, In: Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016, Cham: Springer, 2017. https://doi.org/10.1007/978-3-319-67202-1_1
[22]	J. Wang, X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), 103–115. https://doi.org/10.1016/j.cam.2012.10.003 doi: 10.1016/j.cam.2012.10.003
[23]	L. Mu, J. Wang, Y. Wang, X. Ye, A computational study of the weak Galerkin method for second-order elliptic equations, Numer. Algor., 63 (2013), 753–777. https://doi.org/10.1007/s11075-012-9651-1 doi: 10.1007/s11075-012-9651-1
[24]	Q. L. Li, J. Wang, Weak Galerkin finite element methods for parabolic equations, Numer. Methods Partial Differ. Equ., 29 (2013), 2004–2024. https://doi.org/10.1002/num.21786 doi: 10.1002/num.21786
[25]	L. Mu, J. Wang, X. Ye, S. Zhao, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput., 65 (2015), 363–386. https://doi.org/10.1007/s10915-014-9964-4 doi: 10.1007/s10915-014-9964-4
[26]	J. Wang, X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155–174. https://doi.org/10.1007/s10444-015-9415-2 doi: 10.1007/s10444-015-9415-2
[27]	L. Mu, J. Wang, X. Ye, S. Zhao, A new weak Galerkin finite element method for elliptic interface problems, J. Comput. Phys., 325 (2016), 157–173. https://doi.org/10.1016/j.jcp.2016.08.024 doi: 10.1016/j.jcp.2016.08.024
[28]	S. Toprakseven, A weak Galerkin finite element method on temporal graded meshes for the multi-term time fractional diffusion equations, Comput. Math. Appl., 128 (2022), 108–120. https://doi.org/10.1016/j.camwa.2022.10.012 doi: 10.1016/j.camwa.2022.10.012
[29]	R. Lin, X. Ye, S. Zhang, P. Zhu, A weak Galerkin finite element method for singularly perturbed convection-diffusion-reaction problems, SIAM J. Numer. Anal., 56 (2018), 1482–1497. https://doi.org/10.1137/17M1152528 doi: 10.1137/17M1152528
[30]	P. Zhu, S. Xie, A uniformly convergent weak Galerkin finite element method on Shishkin mesh for 1d convection-diffusion problem, J. Sci. Comput., 85 (2020), 34. https://doi.org/10.1007/s10915-020-01345-3 doi: 10.1007/s10915-020-01345-3
[31]	J. Zhang, X. Liu, Uniform convergence of a weak Galerkin finite element method on Shishkin mesh for singularly perturbed convection-diffusion problems in 2D, Appl. Math. Comput., 432 (2022), 127346. https://doi.org/10.1016/j.amc.2022.127346 doi: 10.1016/j.amc.2022.127346
[32]	S. Toprakseven, P. Zhu, Uniform convergent modified weak Galerkin method for convection- dominated two-point boundary value problems, Turkish J. Math., 45 (2021), 2703–2730. https://doi.org/10.3906/mat-2106-102 doi: 10.3906/mat-2106-102
[33]	S. Toprakseven, P. Zhu, Error analysis of a weak Galerkin finite element method for two- parameter singularly perturbed differential equations in the energy and balanced norms, Appl. Math. Comput., 441 (2023), 127683. https://doi.org/10.1016/j.amc.2022.127683 doi: 10.1016/j.amc.2022.127683
[34]	J. Zhang, X. Liu, Uniform convergence of a weak Galerkin method for singularly perturbed convection-diffusion problems, Math. Comput. Simulation, 200 (2022), 393–403. https://doi.org/10.1016/j.matcom.2022.04.023 doi: 10.1016/j.matcom.2022.04.023
[35]	S. Toprakseven, Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems, Hacet. J. Math. Stat., 52 (2023), 1–26. https://doi.org/10.15672/hujms.1117320 doi: 10.15672/hujms.1117320
[36]	S. Toprakseven, Optimal order uniform convergence in energy and balanced norms of weak Galerkin finite element method on Bakhvalov-type meshes for nonlinear singularly perturbed problems, Comput. Appl. Math., 41 (2022), 377. https://doi.org/10.1007/s40314-022-02090-z doi: 10.1007/s40314-022-02090-z
[37]	X. Liu, J. Zhang, Supercloseness of weak Galerkin method on Bakhvalov-type mesh for a singularly perturbed problem in 1D, Numer. Algorithms, 93 (2023), 367–395. https://doi.org/10.1007/s11075-022-01420-w doi: 10.1007/s11075-022-01420-w
[38]	S. Toprakseven, Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems, Calcolo, 59 (2022), https://doi.org/10.1007/s10092-021-00449-y
[39]	S. Toprakseven, P. Zhu, A parameter-uniform weak Galerkin finite element method for a coupled system of singularly perturbed reaction-diffusion equations, Filomat, 37 (2023), 4351–4374. https://doi.org/10.2298/FIL2313351T doi: 10.2298/FIL2313351T
[40]	T. Linss, M. Stynes, Numerical solution of system of singularly perturbed differential equations, Comput. Methods Appl. Math., 9 (2009), 165–191. https://doi.org/10.2478/cmam-2009-0010 doi: 10.2478/cmam-2009-0010
[41]	T. Linss, Analysis of a FEM for a coupled system of singularly perturbed reaction-diffusion equations, Numer. Algor., 50 (2009), 283–291. https://doi.org/10.1007/s11075-008-9228-1 doi: 10.1007/s11075-008-9228-1
[42]	C. Clavero, J. L. Gracia, F. J. Lisbona, An almost third order finite difference scheme for singularly perturbed reaction-diffusion systems, J. Comput. Appl. Math., 234 (2010), 2501–2515. https://doi.org/10.1016/j.cam.2010.03.011 doi: 10.1016/j.cam.2010.03.011
[43]	S. Toprakseven, A weak Galerkin finite element method for time fractional reaction-diffusion- convection problems with variable coefficients, Appl. Numer. Math., 168 (2021), 1–12. https://doi.org/10.1016/j.apnum.2021.05.021 doi: 10.1016/j.apnum.2021.05.021
[44]	P. Oswald, $L^\infty$-bounds for the $L^2$-projection onto linear spline spaces, In: Recent Advances in Harmonic Analysis and Applications, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-4565-4_24
[45]	T. Zhang, L. Tang, A weak finite element method for elliptic problems in one space dimension, Appl. Math. Comput., 280 (2016), 1–10. https://doi.org/10.1016/j.amc.2016.01.018 doi: 10.1016/j.amc.2016.01.018

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