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A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle

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Abstract

In the present paper, we solve the steady state diffusion equation in 3D domains by means of a cell-centered finite volume method that uses a Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear defect correction strategy (MPFA-DNL) to guarantee the Discrete Maximum Principle (DMP). Our formulation is based in the fact that the flux of MPFA methods can be split into two different parts: a Two Point Flux Approximation (TPFA) component and the Cross-Diffusion Terms (CDT). In the linear MPFA-D method, this split is particularly simple since it lies at the core of the original method construction. In this context, we introduce a non-linear defect correction, aiming to mitigate, whenever necessary, the contributions from the CDT, avoiding, this way, spurious oscillations and DMP violations. Our new MPFA-DNL scheme is locally conservative and capable of dealing with arbitrary anisotropic diffusion tensors and unstructured meshes, without harming the second order convergence rates of the original MPFA-D. To appraise the accuracy and robustness of our formulation, we solve some benchmark problems found in literature. In this paper, we restrict ourselves to tetrahedral meshes, even though, in principle, there is no restriction to extend the method to other polyhedral control volumes.

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Data Availability

Enquiries about data availability should be directed to the authors.

Notes

  1. The meshes we used can be found at the following address: https://github.com/tuliocavalcante/mesh.

References

  1. Keilegavlen, E., Aavatsmark, I.: Monotonicity for MPFA methods on triangular grids. Comput. Geosci. 15, 3–16 (2011). https://doi.org/10.1007/s10596-010-9191-5

    Article  MATH  Google Scholar 

  2. Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I derivation of the methods. SIAM J. Sci. Comput. 19, 1700–1716 (1998). https://doi.org/10.1137/S1064827595293582

    Article  MathSciNet  MATH  Google Scholar 

  3. Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II. Discussion and numerical results. SIAM J. Sci. Comput. 19, 1717–1736 (1998). https://doi.org/10.1137/S1064827595293594

    Article  MathSciNet  MATH  Google Scholar 

  4. Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259–290 (1998). https://doi.org/10.1023/A:1011510505406

    Article  MathSciNet  MATH  Google Scholar 

  5. Fletcher, C.A.J.: Computational Galerkin Methods. Springer, Berlin (1984). https://doi.org/10.1007/978-3-642-85949-6〹

    Book  MATH  Google Scholar 

  6. Ciarlet, P.G.: The finite element method for elliptic problems. Society for Industrial and Applied Mathematics; 2002. https://doi.org/10.1137/1.9780898719208.

  7. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. Gall. I., Magenes E. Math. Asp. Finite Elem. Methods. Lect. Notes Math. vol 606., Springer, Berlin, 1977, pp. 292–315. https://doi.org/10.1007/BFb0064470.

  8. Durán, R.G.: Mixed finite element methods. Boffi D., Gastaldi L. Mix. Finite Elem. Compat. Cond. Appl. Lect. Notes Math. vol 1939., Springer, Berlin, 2008, pp. 1–44. https://doi.org/10.1007/978-3-540-78319-0_1

  9. de Carvalho, D.K.E., Willmersdorf, R.B., Lyra, P.R.M.: Some results on the accuracy of an edge-based finite volume formulation for the solution of elliptic problems in non-homogeneous and non-isotropic media. Int. J. Numer. Methods Fluids 61, 237–254 (2009). https://doi.org/10.1002/fld.1948

    Article  MathSciNet  MATH  Google Scholar 

  10. Ciarlet, P.G., Raviart, P.-A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2, 17–31 (1973). https://doi.org/10.1016/0045-7825(73)90019-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Korotov, S., Křížek, M., Neittaanmäki, P.: Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle. Math. Comput. 70, 107–120 (2000). https://doi.org/10.1090/S0025-5718-00-01270-9

    Article  MathSciNet  MATH  Google Scholar 

  12. Burman, E., Ern, A.: Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes. Comptes Rendus Math. 338, 641–646 (2004). https://doi.org/10.1016/j.crma.2004.02.010

    Article  MathSciNet  MATH  Google Scholar 

  13. Le Potier, C.: A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators. Int. J. Finite Vol Inst Mathématiques Marseille, AMU, pp. 1–20 (2009)

  14. Cancès, C., Cathala, M., Le Potier, C.: Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125, 387–417 (2013). https://doi.org/10.1007/s00211-013-0545-5

    Article  MathSciNet  MATH  Google Scholar 

  15. Pal, M., Edwards, M.G.: Flux-splitting schemes for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. Eur. Conf. Comput. Fluid Dyn. (2006).

  16. Pal, M., Edwards, M.G.: Non-linear flux-splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients. Int. J. Numer. Methods Fluids 66, 299–323 (2011). https://doi.org/10.1002/fld.2258

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, Q.-Y., Wan, J., Yang, Y., Mifflin, R.T.: Enriched multi-point flux approximation for general grids. J. Comput. Phys. 227, 1701–1721 (2008). https://doi.org/10.1016/j.jcp.2007.09.021

    Article  MathSciNet  MATH  Google Scholar 

  18. Kuzmin, D., Shashkov, M.J., Svyatskiy, D.: A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems. J. Comput. Phys. 228, 3448–3463 (2009). https://doi.org/10.1016/j.jcp.2009.01.031

    Article  MathSciNet  MATH  Google Scholar 

  19. 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). https://doi.org/10.1016/j.jcp.2018.06.052

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhao, F., Sheng, Z., Yuan, G.: A monotone combination scheme of diffusion equations on polygonal meshes. ZAMM: J. Appl. Math. Mech./Zeitschrift Für Angew Math Und Mech (2020). https://doi.org/10.1002/zamm.201900320

    Article  Google Scholar 

  21. Herbin, R., Hubert, F.: Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids. Finite Vol. complex Appl. V, pp. 659–692 (2008).

  22. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin Methods for Elliptic Problems, pp. 89–101 (2000). https://doi.org/10.1007/978-3-642-59721-3_5.

  23. Gao, Z., Wu, J.: A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes. Int. J. Numer. Methods Fluids 67, 2157–2183 (2011). https://doi.org/10.1002/fld.2496

    Article  MathSciNet  MATH  Google Scholar 

  24. Contreras, F.R.L., Lyra, P.R.M., Souza, M.R.A., Carvalho, D.K.E.: A cell-centered multipoint flux approximation method with a diamond stencil coupled with a higher order finite volume method for the simulation of oil–water displacements in heterogeneous and anisotropic petroleum reservoirs. Comput. Fluids 127, 1–16 (2016). https://doi.org/10.1016/j.compfluid.2015.11.013

    Article  MathSciNet  MATH  Google Scholar 

  25. Cavalcante, TdeM., Contreras, F.R.L., Lyra, P.R.M., Carvalho, D.K.E.: A multipoint flux approximation with diamond stencil finite volume scheme for the two-dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid-grid method. Int. J. Numer. Methods Fluids (2020). https://doi.org/10.1002/fld.4829

    Article  MathSciNet  Google Scholar 

  26. Lira Filho, R.J.M., Santos, S.R., Cavalcante, TdeM., Contreras, F.R.L., Lyra, P.R.M., Carvalho, D.K.E.: A linearity-preserving finite volume scheme with a diamond stencil for the simulation of anisotropic and highly heterogeneous diffusion problems using tetrahedral meshes. Comput Struct 250, 106510 (2021). https://doi.org/10.1016/j.compstruc.2021.106510

    Article  Google Scholar 

  27. Sheng, Z., Yuan, G.: Construction of Nonlinear Weighted Method for Finite Volume Schemes Preserving Maximum Principle. SIAM J Sci Comput 40, A607–A628 (2018). https://doi.org/10.1137/16M1098000

    Article  MathSciNet  MATH  Google Scholar 

  28. Sheng, Z., Yuan, G., Yue, J.: A nonlinear convex combination in the construction of finite volume scheme satisfying maximum principle. Appl. Numer. Math. 156, 125–139 (2020). https://doi.org/10.1016/j.apnum.2020.04.014

    Article  MathSciNet  MATH  Google Scholar 

  29. Edwards, M.G.: M-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids. J. Comput. Phys. 160, 1–28 (2000). https://doi.org/10.1006/jcph.2000.6418

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhou, H., Sheng, Z., Yuan, G.: A finite volume method preserving maximum principle for the diffusion equations with imperfect interface. Appl. Numer. Math. 158, 314–335 (2020). https://doi.org/10.1016/j.apnum.2020.08.008

    Article  MathSciNet  MATH  Google Scholar 

  31. Véron, L.: Elliptic Equations Involving Measures, pp. 593–712 (2004). https://doi.org/10.1016/S1874-5733(04)80010-X.

  32. Borsuk, M., Kondratiev, V.: The Dirichlet problem for elliptic linear divergent equations in a nonsmooth domain, 2006, pp. 165–213. https://doi.org/10.1016/S0924-6509(06)80018-8.

  33. Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ 24, 1329–1360 (2008). https://doi.org/10.1002/num.20320

    Article  MathSciNet  MATH  Google Scholar 

  34. Møyner, O., Lie, K.-A.: A multiscale two-point flux-approximation method. J. Comput. Phys. 275, 273–293 (2014). https://doi.org/10.1016/j.jcp.2014.07.003

    Article  MathSciNet  MATH  Google Scholar 

  35. Eymard, R., Henry, G., Herbin, R., Hubert, F., Klöfkorn, R., Manzini, G.: 3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids, 2011, pp. 895–930. https://doi.org/10.1007/978-3-642-20671-9_89.

  36. Queiroz, L.E.S., Souza, M.R.A., Contreras, F.R.L., Lyra, P.R.M., de Carvalho, D.K.E.: On the accuracy of a nonlinear finite volume method for the solution of diffusion problems using different interpolations strategies. Int. J. Numer. Methods Fluids 74, 270–291 (2014). https://doi.org/10.1002/fld.3850

    Article  MathSciNet  MATH  Google Scholar 

  37. Danilov, A.A., Vassilevski, Y.V.: A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russ J. Numer. Anal. Math. Model (2009). https://doi.org/10.1515/RJNAMM.2009.014

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the Foundation for Support of Science and Technology of Pernambuco (FACEPE), the National Council for Scientific Development (CNPq) and the Coordination for Improvement of Higher Education Personnel (CAPES).

Funding

The authors received financial support from some research funding agencies during the preparation of this manuscript, namely the Brazilian Research Council - CNPq (PQ-308334/2019-1, PQ-310242/2018-5 and PQ-310145/2021-0), Foundation for the Support of Science and Technology of Pernambuco - FACEPE (IBPG-1160-3.01/18 and IBPG-0017-3.01/18), and FADE / UFPE / Energi Simulation (Funding No. 53/2020).

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Authors and Affiliations

  1. Department of Civil Engineering, Institute for Petroleum and Energy Research, UFPE, Room 541, Av. Arquitetura s/n, Recife, PE, 50740-550, Brazil

    T. M. Cavalcante, R. J. M. Lira Filho & A. C. R. Souza

  2. Department of Mechanical Engineering, UFPE, Recife, PE, Brazil

    D. K. E. Carvalho & P. R. M. Lyra

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  1. T. M. Cavalcante
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Correspondence to T. M. Cavalcante.

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Cavalcante, T.M., Filho, R.J.M.L., Souza, A.C.R. et al. A Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear Defect Correction Strategy for the Numerical Solution of Steady State Diffusion Problems in Heterogeneous and Anisotropic Media Satisfying the Discrete Maximum Principle. J Sci Comput 93, 42 (2022). https://doi.org/10.1007/s10915-022-01978-6

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