Abstract
We propose and analyze a family of decoupled and positivity-preserving discrete duality finite volume (DDFV) schemes for diffusion problems on general polygonal meshes. These schemes are further extensions of the scheme in Su et al. (J. Comput. Phys.372, 773–798, 2018) and share some benefits. First, the two sets of finite volume (FV) equations are constructed for the vertex-centered and cell-centered unknowns, respectively, and these two sets of equations can be solved in a decoupled way. Second, unlike most existing nonlinear positivity-preserving schemes, nonlinear iteration methods are not required for linear problems and the nonlinear solver can be selected unrestrictedly for nonlinear problems. Moreover, the positivity and well-posedness of the solution can be analyzed rigorously for linear problems. Under some weak geometric assumptions, the stability and error estimate results for the vertex-centered unknowns are obtained. Since the present DDFV schemes and the pure vertex-centered scheme in Wu et al. (Int. J. Numer. Meth. Fluids81(3), 131–150, 2016) share the same FV equations for the vertex-centered unknowns, this part itself can also serve as an analysis for the scheme in Wu et al. (Int. J. Numer. Meth. Fluids81(3), 131–150, 2016). Furthermore, by assuming the coercivity of the FV equations for the cell-centered unknowns, a first-order H1 error estimate is obtained for the cell-centered unknowns through the analysis of residual errors. Numerical experiments demonstrate both the accuracy and the positivity of discrete solutions on general grids. The efficiency of the Newton method is emphasized by comparison with the fixed-point method, which illustrates the advantage of our schemes when dealing with nonlinear problems.
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References
Adams, R.A.: Sobolov Spaces. Academic Press, New York (1975)
Agelas, L., Pietro, D., Droniou, J.: The G method for heterogeneous anisotropic diffusion on general meshes. ESAIM:M2AN 44, 597–625 (2010)
Blanc, X., Labourasse, E.: A positive scheme for diffusion problems on deformed meshes. Z. Angew. Math. Mech. 96(6), 660–680 (2016)
Bonelle, J., Ern, A.: Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM:M2AN 48(2), 553–581 (2014)
Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. ESAIM:M2AN 43(2), 277–295 (2009)
Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15, 1533–1551 (2005)
Camier, J., Hermeline, F.: A monotone nonlinear finite volume method for approximating diffusion operators on general meshes. Int. J. Numer. Meth. Engng. 107(6), 496–519 (2016)
Cancès, C., Cathala, M., Le Potier, C.: Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125(3), 387–417 (2013)
Cancès, C., Guichard, C.: Convergence of a nonlinear entropy diminishing control volume finite element scheme for solving anisotropic degenerate parabolic equations. Math. Comp. 85(298), 549–580 (2016)
Coudiére, Y., Vila, J., Villedieu, P.: Convergence rate of a finite volume scheme for a two-dimensional diffusion convection problem. ESAIM:M2AN 33, 493–516 (1999)
Després, B.: Non linear schemes for the heat equation in 1d. ESAIM: M2AN 48(1), 107–134 (2014)
Droniou, J., Le Potier, C.: Construction and convergence study of schemes preserving the elliptic local maximum principle. SIAM J. Numer. Anal. 49(2), 459–490 (2011)
Domelevo, K., Omnes, P.: A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM-math. Model. Num. Anal. 39(6), 1203–1249 (2005)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, vol. VII, Handb. Numer. Anal., VII, North-Holland Amsterdam, pp 713–1020 (2000)
Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Finite Volumes for Complex Applications V, pp 659–692. Wiley, New York (2008)
Hermeline, F.: Une méthode de volumes finis pour les équations elliptiques du second ordre. C. R. Acad. Sci. Paris Ser. I Math. 326(12), 1433–1436 (1998)
Hermeline, F.: A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160(2), 481–499 (2000)
Huang, W., Kappen, A.M.: A Study of Cell-Center Finite Volume Methods for Diffusion Equations. University of Kansas, Mathematics Research Report (1998)
Keilegavlen, E., Nordbotten, J.M., Aavatsmark, I.: Sufficient criteria are necessary for monotone control volume methods. Appl. Math. Lett. 22(8), 1178–1180 (2009)
Le Potier, C.: Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés. C. R. Acad. Sci. Paris, Ser. I 340(12), 787–792 (2005)
Le Potier, C., Mahamane, A.: A nonlinear correction and maximum principle for diffusion operators with hybrid schemes. C. R. Acad. Sci. Paris, Ser. I 350, 101–106 (2012)
Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Y.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 227(1), 492–512 (2007)
Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes. J. Comput. Phys. 228(3), 703–716 (2009)
Nordbotten, J.M., Aavatsmark, I., Eigestad, G.T.: Monotonicity of control volume methods. Numer. Math. 106(2), 255–288 (2007)
Le Potier, C.: A nonlinear correction and local minimum principle for diffusion operators with finite differences. C. R. Acad. Sci. Paris, Ser. I 356(1), 100–106 (2018)
Schnerder, M., Agélas, L., Enchéry, G., Flemisch, B.: Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. J. Comput. Phys. 351, 80–107 (2017)
Su, S., Dong, Q., Wu, J.: A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes. J. Comput. Phys. 372, 773–798 (2018)
Wu, J.: Vertex-centered linearity-preserving schemes for nonlinear parabolic problems on polygonal grids. J. Sci. Comput. 71, 499–524 (2017)
Wu, J., Dai, Z., Gao, Z., Yuan, G.: Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes. J. Comput. Phys. 229, 3382–3401 (2010)
Wu, J., Gao, Z., Dai, Z.: A vertex-centered linearity-preserving discretization of diffusion problems on polygonal meshes. Int. J. Numer. Meth. Fluids 81(3), 131–150 (2016)
Yin, L., Wu, J., Gao, Z.: The cell functional minimization scheme for the anisotropic diffusion problems on arbitrary polygonal grids. ESAIM: M2AN 49(1), 193–220 (2015)
Zhang, X., Su, S., Wu, J.: A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids. J. Comput. Phys. 344, 419–436 (2017)
Acknowledgements
The authors would like to thank the reviewers for their careful readings and useful suggestions.
Funding
This work was partially supported by the National Natural Science Foundation of China (No. 11871009), China Postdoctoral Science Foundation (No. BX20190013), and CAEP Foundation (No. CX2019028).
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Communicated by: Aihui Zhou
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Dong, Q., Su, S. & Wu, J. Analysis of the decoupled and positivity-preserving DDFV schemes for diffusion problems on polygonal meshes. Adv Comput Math 46, 12 (2020). https://doi.org/10.1007/s10444-020-09748-4
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DOI: https://doi.org/10.1007/s10444-020-09748-4