×

A novel cell-centered approach of upwind types for convection diffusion equations on general meshes. (English) Zbl 07446686

Summary: In this paper, we present novel cell-centered finite element methods for the convection-dominated convection-diffusion problems on the general meshes. The proposed schemes can be constructed from a general mesh by building with a dual mesh and its triangular sub-mesh. Moreover, the schemes are based on piecewise linear functions combined with two upwind techniques on the dual sub-mesh in order to stabilize the numerical solutions and eliminate the spurious oscillations. Through numerical results, two methods are shown to be effective in terms of accuracy, stability and computational cost.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76R05 Forced convection
76R50 Diffusion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aavatsmark, I., Barkve, T.Boe, O. and Mannseth, T. [1998a] “ Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: derivation of the methods,” SIAM J. Sci. Comput.19, 1700-1716. · Zbl 0951.65080
[2] Aavatsmark, I., Barkve, T.Boe, O. and Mannseth, T. [1998a] “ Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: discussion and numerical results,” SIAM J. Sci. Comput.19, 1717-1736. · Zbl 0951.65082
[3] Almeida, R. C. and Silva, R. S. [1997] “ A stable Petrov-Galerkin method for convection-dominated problems,” Comput. Methods Appl. Mech. Eng.140, 291-304. · Zbl 0899.76258
[4] Bertolazzi, E. and Manzini, G. [2004] “ A cell-centered second-order accurate finite volume method for convection-diffusion problems on unstructured meshes,” M3AS14, 1235-1260. · Zbl 1079.65113
[5] Brezzi, F., Buffa, A. and Lipnikov, K. [2009] “ Mimetic finite differences for elliptic problems,” Math. Model Numer. Anal.43, 277-295. · Zbl 1177.65164
[6] Brooks, A. and Hughes, T. [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, 199-259. · Zbl 0497.76041
[7] Chainais-Hillairet, C. and Droniou, J. [2009] “ Finite volume schemes for non-coercive elliptic problems with neumann boundary conditions,” IMA J. Numer. Anal.31, 61-85. · Zbl 1211.65144
[8] Ciarlet, P. G. and Lions J. L. (eds.) [1991] “Basic error estimates for elliptic problems”, Handbook of Numerical Analysis, Finite Element Methods (pt. 1), pp. 17-351. · Zbl 0875.65086
[9] Dawson, C. and Aizinger, V. [1999] “ Upwind-mixed methods for transport equations,” Comput. Geosci.3, 93-110. · Zbl 0962.65084
[10] Droniou, J. and Eymard, R. [2006] “ A mixed finite volume scheme for anisotropic diffusion problems on any grids,” Numer. Math.105, 35-71. · Zbl 1109.65099
[11] Droniou, J., Eymard, R. and Herbin, R. [2010] “ A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods,” Math. Models Methods Appl. Sci.20, 265-295. · Zbl 1191.65142
[12] Heinrich, J., Huyakorn, P., Zienkiewicz, O. and Mitchell, A. [1977] “ An ‘upwind’ finite element scheme for two-dimensional convective transport equation,” Int. J. Numer. Method Eng.11, 131-143. · Zbl 0353.65065
[13] Heinrich, J. and Zienkiewicz, O. [1997] “ Quadratic finite element schemes for two-dimensional convective-transport problems,” Int. J. Numer. Method Eng.11, 1831-1844. · Zbl 0372.76002
[14] Jaffre, J. [1984] “ Decentrage et elements finis mixtes pour les equations de diffusion-convection,” Calcolo21, 171-197. · Zbl 0562.65077
[15] John, V. and Knobloch, P. [2007] “ On spurious oscillations at layer diminishing (SOLD) methods for convection-diffusion equations: Part I — A review,” Comput. Methods Appl. Mech. Eng.196, 2197-2215. · Zbl 1173.76342
[16] Knobloch, P. [2006] “ Improvement of the Mizukami-Hughes method for convection-diffusion equations,” Comput. Methods Appl. Mech. Eng.196, 579-594. · Zbl 1120.76328
[17] Knobloch, P. [2006] “ Application of the Mizukami-Hughes method to bilinear finite elements”, Czech-Jpn. Seminar Appl. Math.137-147.
[18] Knobloch, P. [2008] “ Numerical solution of convection-diffusion equations using a nonlinear method of upwind type,” J. Sci. Comput.43, 454-470. · Zbl 1203.76084
[19] Matthies, G. and Tobiska, L. [2001] “ The streamline-diffusion method for conforming and nonconforming finite elements of lowest order applied to convection-diffusion problems,” Computing66, 343-364. · Zbl 0998.65118
[20] Mizukami, A. and Hughes, T. J. [1985] “ A petrov-galerkin finite element method for convection-dominated flows: An accurate upwinding technique for satisfying the maximum principle,” Comput. Methods Appl. Mech. Eng.50, 181-193. · Zbl 0553.76075
[21] Le Potier, C. and Ong, T. H. [2012] “ A cell-centered scheme for heterogeneous anisotropic diffusion problems on general meshes”, Int. J. Finite Vol.8, 64-77.
[22] Roos, H. G., Stynes, M. and Tobiska, L. [1996], “ Numerical methods for singularly perturbed differential equations”, Convection-Diffusion and Flow Problems (Springer). · Zbl 0844.65075
[23] Scharfetter, D. L. and Gummel, H. K. [1969] “ Large signal analysis of a silicon read diode,” IEEE Trans. Electron Dev.16, 64-77.
[24] Tabata, M. [1977] “ A finite element approximation corresponding to the upwind finite differencing,” Mem. Numer. Math.4, 47-63. · Zbl 0358.65102
[25] Veiga, B. D., Droniou, J. and Manzini, G. [2011] “ A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems,” IMA J. Numer. Anal.31, 1357-1401. · Zbl 1263.65102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.