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A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

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Abstract

In this paper, we design and develop a weak Galerkin finite-element numerical method for solving singularly perturbed convection–diffusion–reaction equations with a non-conservation convection term. Many advantages of the proposed method include support for the higher order of convergence and general polygonal meshes. The convergence study for the weak Galerkin algorithm is performed in both the triple-bar norm and the \(L^2\) norm. We achieve an optimal order of convergence of \(\mathcal{O}(h^k)\) in the triple-bar norm and \(\mathcal{O}(h^{k+1})\) in the \(L^2\) norm. Several numerical experiments in a two-dimensional setting are carried out to demonstrate the convergence of our theories.

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References

  • Adams R, Fournier J (2003) Sobolev spaces, 2nd edn. Academic Press, Amsterdam

    Google Scholar 

  • Ayuso B, Marini LD (2009) Discontinuous Galerkin methods for advection–diffusion–reaction problems. SIAM J Numer Anal 47(2):1391–1420

    Article  MathSciNet  Google Scholar 

  • Baumann CE, Oden JT (1999) A discontinuous hp finite element method for convection–diffusion problems. Comput Methods Appl Mech Eng 175(3–4):311–341

    Article  MathSciNet  ADS  Google Scholar 

  • Brooks A (1991) Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 87:364–384

    Google Scholar 

  • Buffa A, Hughes TJ, Sangalli G (2006) Analysis of a multiscale discontinuous Galerkin method for convection–diffusion problems. SIAM J Numer Anal 44(4):1420–1440

    Article  MathSciNet  Google Scholar 

  • Burman E (2010) Consistent supg-method for transient transport problems: stability and convergence. Comput Methods Appl Mech Eng 199(17–20):1114–1123

    Article  MathSciNet  ADS  Google Scholar 

  • Burman E, Ern A (2007) Continuous interior penalty hp-finite element methods for advection and advection–diffusion equations. Math Comput 76(259):1119–1140

    Article  MathSciNet  ADS  Google Scholar 

  • Chen G, Feng M, Xie X (2017) A robust WG finite element method for convection–diffusion–reaction equations. J Comput Appl Math 315:107–125

    Article  MathSciNet  Google Scholar 

  • Deka B, Kumar N (2021) Error estimates in weak Galerkin finite element methods for parabolic equations under low regularity assumptions. Appl Numer Math 162:81–105

    Article  MathSciNet  Google Scholar 

  • Deka B, Kumar N (2021b) A systematic study on weak Galerkin finite element method for second order parabolic problems. arXiv preprint arXiv:2103.13669

  • Farrell P, Hegarty A, Miller JM, O’Riordan E, Shishkin GI (2000) Robust computational techniques for boundary layers. Chapman and Hall/CRC, Boca Raton

    Book  Google Scholar 

  • Gharibi Z, Dehghan M (2021) Convergence analysis of weak Galerkin flux-based mixed finite element method for solving singularly perturbed convection-diffusion-reaction problem. Appl Numer Math 163:303–316

    Article  MathSciNet  Google Scholar 

  • Guzmán J (2006) Local analysis of discontinuous Galerkin methods applied to singularly perturbed problems

  • Houston P, Schwab C, Süli E (2002) Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J Numer Anal 39(6):2133–2163

    Article  MathSciNet  Google Scholar 

  • Huang Y, Li J, Li D (2017) Developing weak Galerkin finite element methods for the wave equation. Numer Methods Partial Differ Equ 33(3):868–884

    Article  MathSciNet  Google Scholar 

  • John V, Novo J (2011) Error analysis of the supg finite element discretization of evolutionary convection–diffusion–reaction equations. SIAM J Numer Anal 49(3):1149–1176

    Article  MathSciNet  Google Scholar 

  • Li QH, Wang J (2013) Weak Galerkin finite element methods for parabolic equations. Numer Methods Partial Differ Equ 29(6):2004–2024

    Article  MathSciNet  Google Scholar 

  • Lin R, Stynes M (2012) A balanced finite element method for singularly perturbed reaction–diffusion problems. SIAM J Numer Anal 50(5):2729–2743

    Article  MathSciNet  Google Scholar 

  • Lin R, Ye X, Zhang S, Zhu P (2018) A weak Galerkin finite element method for singularly perturbed convection–diffusion–reaction problems. SIAM J Numer Anal 56(3):1482–1497

    Article  MathSciNet  Google Scholar 

  • Lin R, Ye X, Zhang S, Zhu P (2018) A weak Galerkin finite element method for singularly perturbed convection–diffusion–reaction problems. SIAM J Numer Anal 56(3):1482–1497

    Article  MathSciNet  Google Scholar 

  • Linss T, Roos H-G, Vulanovic R (2000) Uniform pointwise convergence on shishkin-type meshes for quasi-linear convection–diffusion problems. SIAM J Numer Anal 38(3):897–912

    Article  MathSciNet  Google Scholar 

  • Morton KW (2019) Numerical solution of convection–diffusion problems. CRC Press, Boca Raton

    Book  Google Scholar 

  • Mu L, Chen Z (2021) A new WENO weak Galerkin finite element method for time dependent hyperbolic equations. Appl Numer Math 159:106–124

    Article  MathSciNet  Google Scholar 

  • Mu L, Ye JWX (2015) Weak Galerkin finite element methods on polytopal meshes. Int J Numer Anal Model 12(1):31–53

    MathSciNet  ADS  Google Scholar 

  • Mu L, Wang J, Ye X (2017) A least-squares-based weak Galerkin finite element method for second order elliptic equations. SIAM J Sci Comput 39(4):A1531–A1557

    Article  MathSciNet  Google Scholar 

  • Roos H-G, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, vol 24. Springer, Berlin

    Google Scholar 

  • Schieweck F (2008) On the role of boundary conditions for cip stabilization of higher order finite elements. Electron Trans Numer Anal 32(1–16):62

    MathSciNet  Google Scholar 

  • Sharma N (2021) Robust a-posteriori error estimates for weak Galerkin method for the convection–diffusion problem. Appl Numer Math 170:384–397

    Article  MathSciNet  Google Scholar 

  • Stynes M, O’Riordan E (1997) A uniformly convergent Galerkin method on a shishkin mesh for a convection–diffusion problem. J Math Anal Appl 214(1):36–54

    Article  MathSciNet  Google Scholar 

  • Talischi C, Paulino GH, Pereira A, Menezes IF (2012) Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45(3):309–328

    Article  MathSciNet  Google Scholar 

  • Tobiska L (2006) Analysis of a new stabilized higher order finite element method for advection–diffusion equations. Comput Methods Appl Mech Eng 196(1–3):538–550

    Article  MathSciNet  ADS  Google Scholar 

  • Toprakseven Ş (2021) A weak Galerkin finite element method for time fractional reaction–diffusion–convection problems with variable coefficients. Appl Numer Math 168:1–12

    Article  MathSciNet  Google Scholar 

  • Toprakseven S (2022) Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems. Calcolo 59:1–35

    Article  MathSciNet  Google Scholar 

  • Toprakseven Ş (2022) A weak Galerkin finite element method on temporal graded meshes for the multi-term time fractional diffusion equations. Comput Math Appl 128:108–120

    Article  MathSciNet  Google Scholar 

  • Toprakseven Ş, Zhu P (2023) 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 441:127683

    MathSciNet  Google Scholar 

  • Wang J, Ye X (2013) A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math 241:103–115

    Article  MathSciNet  Google Scholar 

  • Wang J, Ye X (2014) A weak Galerkin mixed finite element method for second order elliptic problems. Math Comput 83(289):2101–2126

    Article  MathSciNet  Google Scholar 

  • Wang J, Wang R, Zhai Q, Zhang R (2018) A systematic study on weak Galerkin finite element methods for second order elliptic problems. J Sci Comput 74(3):1369–1396

    Article  MathSciNet  Google Scholar 

  • Wang X, Gao F, Sun Z (2020) Weak Galerkin finite element method for viscoelastic wave equations. J Comput Appl Math 375:112816

    Article  MathSciNet  Google Scholar 

  • Wheeler MF (1978) An elliptic collocation-finite element method with interior penalties. SIAM J Numer Anal 15(1):152–161

    Article  MathSciNet  Google Scholar 

  • Xie S, Zhu P, Wang X (2019) Error analysis of weak Galerkin finite element methods for time-dependent convection–diffusion equations. Appl Numer Math 137:19–33

    Article  MathSciNet  Google Scholar 

  • Zhang H, Zou Y, Xu Y, Zhai Q, Yue H (2016) Weak Galerkin finite element method for second order parabolic equations. Int J Numer Anal Model 13(4):525–544

    MathSciNet  Google Scholar 

  • Zhou G (1997) How accurate is the streamline diffusion finite element method? Math Comput 66(217):31–44

    Article  MathSciNet  ADS  Google Scholar 

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Acknowledgements

The work was supported by the Indian Institute of Technology Roorkee, India, with grant No. IITR/Estt. (A)-Rect. Cell/E-5001 (92)/7746. Also, the third author (Ram Jiwari) is thankful to the Science and Engineering Research Board (SERB), India, for supporting Project No. MTR/2021/000059.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Roorkee, Roorkee, 247667, India

    Naresh Kumar & Ram Jiwari

  2. Department of Computer Science, Artvin Çoruh University, 08100, Artvin, Turkey

    Şuayip Toprakseven

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  1. Naresh Kumar
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  2. Şuayip Toprakseven
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  3. Ram Jiwari
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Correspondence to Ram Jiwari.

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Kumar, N., Toprakseven, Ş. & Jiwari, R. A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes. Comp. Appl. Math. 43, 44 (2024). https://doi.org/10.1007/s40314-023-02553-x

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  • DOI: https://doi.org/10.1007/s40314-023-02553-x

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