Abstract
In this paper, a weak Galerkin (WG) finite-element method is developed and analyzed for singularly perturbed convection–diffusion–reaction problems with interface. The WG algorithm is very simple in terms of variational formulation and has less unknowns. This algorithm allows us to use finite-element partitions with general polytopal meshes and is also easily generalizable to higher orders. This method can accommodate very complicated interfaces because of its high flexibility in choosing the finite-element partitions. Optimal order error estimate is established for the WG approximations in discrete \(H^{1}\) norm. Numerical experiments are presented to confirm our theoretical finding and to show efficiency, flexibility, accuracy and robustness of the WG interface approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
This manuscript has no associated data.
References
Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32(1–3):199–259
Cangiani A, Georgoulis EH, Houston P (2014) hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math Models Methods Appl Sci 24(10):2009–2041
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
Ciarlet PG (2002) The finite element method for elliptic problems. SIAM, Philadelphia
Deka B (2018) A weak Galerkin finite element method for elliptic interface problems with polynomial reduction. Numer Math Theory Methods Appl 11(3)
Dong H, Ying W, Zhang J (2018) A hybridizable discontinuous Galerkin method for elliptic interface problems in the formulation of boundary integral equations. J Comput Appl Math 344:624–639
Hughes TJR (1979) A multidimentional upwind scheme with no crosswind diffusion. In: Finite element methods for convection dominated flows, AMD, p 34
Huynh LNT, Nguyen NC, Peraire J, Khoo BC (2013) A high-order hybridizable discontinuous Galerkin method for elliptic interface problems. Int J Numer Meth Eng 93(2):183–200
Khan A, Upadhyay CS, Gerritsma M (2018) Spectral element method for parabolic interface problems. Comput Methods Appl Mech Eng 337:66–94
LeVeque RJ, Zhilin L (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31(4):1019–1044
Li Z, Ito K (2006) The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains. SIAM, Philadelphia
Lin T (2003) Layer-adapted meshes for convection-diffusion problems. Comput Methods Appl Mech Eng 192(9–10):1061–1105
Lin T (2009) Layer-adapted meshes for reaction-convection-diffusion problems. Springer, Berlin
Lin M, Wang J, Ye X, Zhang S (2015) A weak Galerkin finite element method for the Maxwell equations. J Sci Comput 65(1):363–386
Lin M, Wang J, Ye X, Zhao S (2016) A new weak Galerkin finite element method for elliptic interface problems. J Comput Phys 325:157–173
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
Miller JJH, O’riordan E, Shishkin GI (1996) Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions. World scientific, Singapore
Peskin CS (1977) Numerical analysis of blood flow in the heart. J Comput Phys 25(3):220–252
Peskin CS, McQueen DM (1989) A three-dimensional computational method for blood flow in the heart i. immersed elastic fibers in a viscous incompressible fluid. J Comput Phys 81(2):372–405
Roos H-G (1998) Layer-adapted grids for singular perturbation problems. ZAMM J Appl Math Mech/Z Angew Math Mech Appl Math Mech 78(5):291–309
Shishkin GI (1990) Grid approximation of singularly perturbed elliptic and parabolic equations. Second doctorial thesis, Keldysh Institute, Moscow, Russian. Section 19
Wang C, Wang J (2014) An efficient numerical scheme for the Biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput Math Appl 68(12):2314–2330
Wang J, Ye X (2013) A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math 241:103–115
Wang J, Ye X (2014) A weak Galerkin mixed finite element method for second order elliptic problems. Math Comput 83(289):2101–2126
Wang J, Ye X (2016) A weak Galerkin finite element method for the Stokes equations. Adv Comput Math 42(1):155–174
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
Zhang T, Tang L (2016) A weak finite element method for elliptic problems in one space dimension. Appl Math Comput 280:1–10
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors certify that there is no actual or potential conflict of interest in relation to this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ahmed, T., Baruah, R. & Kumar, R. A weak Galerkin finite-element method for singularly perturbed convection–diffusion–reaction problems with interface. Comp. Appl. Math. 42, 318 (2023). https://doi.org/10.1007/s40314-023-02438-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02438-z