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Staggered DG Method with Small Edges for Darcy Flows in Fractured Porous Media

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Abstract

In this paper, we present and analyze a staggered discontinuous Galerkin method for Darcy flows in fractured porous media on fairly general meshes. A staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions are exploited for the bulk region and the fracture, respectively. Our current analysis works on fairly general polygonal elements even in the presence of small edges. We prove the optimal convergence estimates in \(L^2\) error for all the variables by exploiting the Ritz projection. Importantly, our error estimates are shown to be fully robust with respect to the heterogeneity and anisotropy of the permeability coefficients. Several numerical experiments including meshes with small edges and anisotropic meshes are carried out to confirm the theoretical findings. Finally, our method is applied in the framework of unfitted mesh.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flows in porous media. SIAM J. Sci. Comput. 26, 1667–1691 (2006)

    MathSciNet  MATH  Google Scholar 

  2. D’Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM-Math. Model. Numer. Anal. 46, 465–489 (2012)

    MATH  Google Scholar 

  3. Chave, F., Di Pietro, D.A., Formaggia, L.: A hybrid high-order method for Darcy flows in fractured porous media. SIAM J. Sci. Comput. 40, A1063–A1094 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Antonietti, P.F., Facciolà, C., Russo, A., Verani, M.: Discontinuous Galerkin approximation of flows in fractured porous media on polytopic grids. SIAM J. Sci. Comput. 41, A109–A138 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Jaffré, J., Roberts, J.E.: Modeling flow in porous media with fractures; discrete fracture models with matrix-fracture exchange. Numer. Anal. Appl. 5(2), 162–167 (2012)

    MATH  Google Scholar 

  6. Lipnikov, K., Moulton, J.D., Svyatskiy, D.: A multilevel multiscale mimetic (M3) method for two-phase flows in porous media. J. Comput. Phys. 227, 6727–6753 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flows in fractured porous media. ESAIM-Math. Model. Numer. Anal. 43, 239–275 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Sandve, T.H., Berre, I., Nordbotten, J.M.: An efficient multi-point flux approximation method for discrete fracture-matrix simulations. J. Comput. Phys. 231, 3784–3800 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Berrone, S., Pieraccini, S., Scialó, S.: On simulations of discrete fracture network flows with an optimization-based extended finite element method. SIAM J. Sci. Comput. 35, A908–A935 (2013)

    MathSciNet  MATH  Google Scholar 

  10. Fumagalli, A., Scotti, A.: A numerical method for two-phase flow in fractured porous media with non-matching grids. Adv. Water Resour. 62, 454–464 (2013)

    MATH  Google Scholar 

  11. Benedetto, M.F., Berrone, S., Pieraccini, S., Scialò, S.: The virtual element method for discrete fracture network simulations. Comput. Meth. Appl. Mech. Eng. 280, 135–156 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Brenner, K., Groza, M., Guichard, C., Lebeau, G., Masson, R.: Gradient discretization of hybrid dimensional Darcy flows in fractured porous media. Numer. Math. 134, 569–609 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Verzotti, N.: Mimetic finite difference approximation of flows in fractured porous media. ESAIM-Math. Model. Numer. Anal. 50, 809–832 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Chen, H., Salama, A., Sun, S.: Adaptive mixed finite element methods for Darcy flow in fractured porous media. Water Resour. Res. 52, 7851–7868 (2016)

    Google Scholar 

  15. Chen, H., Sun, S.: A residual-based a posteriori error estimator for single-phase Darcy flow in fractured porous media. Numer. Math. 136, 805–839 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Brenner, K., Hennicker, J., Masson, R., Samier, P.: Gradient discretization of hybrid dimensional Darcy flows in fractured porous media with discontinuous pressure at matrix fracture interfaces. IMA J. Numer. Anal. 37, 1551–1585 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Del Pra, M., Fumagalli, A., Scotti, A.: Well posedness of fully coupled fracture/bulk Darcy flow with XFEM. SIAM J. Numer. Anal. 55, 785–811 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Boon, W.M., Nordboteen, J.M., Yotov, I.: Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56, 2203–2233 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Formaggia, L., Scotti, A., Sottocasa, F.: Analysis of a mimetic finite difference approximation of flows in fractured porous media. ESAIM-Math. Model. Numer. Anal. 52, 595–630 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for wave propagation. SIAM J. Numer. Anal. 44, 2131–2158 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Chung, E.T., Engquist, B.: Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820–3848 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Chung, E.T., Kim, H.H., Widlund, O.B.: Two-level overlapping schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51, 47–67 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Chung, E.T., Ciarlet, P., Jr., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids. J. Comput. Phys. 235, 14–31 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Kim, H.H., Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51, 3327–3350 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Cheung, S.W., Chung, E.T., Kim, H.H., Qian, Y.: Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 302, 251–266 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Lee, J.J., Kim, H.H.: Analysis of a staggered discontinuous Galerkin method for linear elasticity. J. Sci. Comput. 66, 625–649 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Chung, E.T., Qiu, W.: Analysis of an SDG method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 55, 543–569 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Chung, E.T., Park, E.-J., Zhao, L.: Guaranteed a posteriori error estimates for a staggered discontinuous Galerkin method. J. Sci. Comput. 75, 1079–1101 (2018)

    MathSciNet  MATH  Google Scholar 

  29. Zhao, L., Park, E.-J.: A priori and a posteriori error analysis of a staggered discontinuous Galerkin method for convection dominant diffusion equations. J. Comput. Appl. Math. 346, 63–83 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Chung, E., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52, 915–932 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Zhao, L., Park, E.-J.: A staggered discontinuous Galerkin method of minimal dimension on quadrilateral and polygonal meshes. SIAM J. Sci. Comput. 40, A2543–A2567 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Zhao, L., Park, E.-J., Shin, D.-W.: A staggered DG method of minimal dimension for the Stokes equations on general meshes. Comput. Meth. Appl. Mech. Eng. 345, 854–875 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Kim, D., Zhao, L., Park, E.-J.: Staggered DG methods for the pseudostress-velocity formulation of the Stokes equations on general meshes. SIAM J. Sci. Comput. 42, A2537–A2560 (2020)

    MathSciNet  MATH  Google Scholar 

  34. Zhao, L., Park, E.-J.: A new hybrid staggered discontinuous Galerkin method on general meshes. J. Sci. Comput. (2020). https://doi.org/10.1007/s10915-019-01119-6

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhao, L., Park, E.-J.: A lowest-order staggered DG method for the coupled Stokes-Darcy problem. IMA J. Numer. Anal. 40, 2871–2897 (2020)

    MathSciNet  MATH  Google Scholar 

  36. Zhao, L., Chung, E.T., Lam, M.F.: A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits. Comput. Meth. Appl. Mech. Eng. 364, 112986 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Zhao, L., Chung, E.T., Park, E.-J., Zhou, G.: Staggered DG method for coupling of the Stokes and Darcy-Forchheimer problems. SIAM J. Numer. Anal. 59, 1–31 (2021)

    MathSciNet  MATH  Google Scholar 

  38. Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P.: hp-version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes. Springer, Cham (2017)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. Di Pietro, D.A., Ern, A.: A Hybrid High-Order locking-free method for linear elasticity on general meshes. Comput. Meth. Appl. Mech. Engrg. 283, 1–21 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Meth. Appl. Sci. 23, 199–214 (2013)

    MathSciNet  MATH  Google Scholar 

  42. Beirao da Veiga, L., Lovadina, C., Russo, A.: Stability analysis for the virtual element method. Math. Models Meth. Appl. Sci. 27, 2557–2594 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Brenner, S.C., Sung, L.-Y.: Virtual element methods on meshes with small edges or faces. Math. Models Meth. Appl. Sci. 28, 1291–1336 (2018)

    MathSciNet  MATH  Google Scholar 

  44. Cao, S., Chen, L.: Anisotropic error estimates of the linear virtual element method on polygonal meshes. SIAM J. Numer. Anal. 56, 2913–2939 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Cao, S., Chen, L.: Anisotropic error estimates of the linear nonconforming virtual element methods. SIAM J. Numer. Anal. 57, 1058–1081 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Cangiani, A., Georgoulis, E.H., Pryer, T., Sutton, O.J.: A posteriori error estimates for the virtual element method. Numer. Math. 137, 857–893 (2017)

    MathSciNet  MATH  Google Scholar 

  47. Guan, Q.: Weak Galerkin finite element method for Poisson’s equation on polytopal meshes with small edges or faces. J. Comput. Appl. Math. 368, 112584 (2020)

    MathSciNet  MATH  Google Scholar 

  48. Chung, E.T., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52, 915–932 (2014)

    MathSciNet  MATH  Google Scholar 

  49. Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  50. Apel, T.: Anisotropic finite elements: local estimates and applications. In: Advances in Numerical Mathematics (1999)

  51. Duan, H.-Y., Tan, R.C.E.: On the Poincaré-Friedrichs inequality for piecewise $H^1$ functions in anisotropic discontinuous Galerkin finite element methods. Math. Comp. 80, 119–140 (2011)

    MathSciNet  MATH  Google Scholar 

  52. Chu, C.-C., Graham, I.G., Hou, T.-Y.: A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comp. 79, 1915–1955 (2010)

    MathSciNet  MATH  Google Scholar 

  53. Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41, 637–676 (1999)

    MathSciNet  MATH  Google Scholar 

  54. Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.M.: PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscip. Optim. 45, 309–328 (2012)

    MathSciNet  MATH  Google Scholar 

  55. Badia, S., Verdugo, F., Martín, A.F.: The aggregated unfitted finite element method for elliptic problems. Comput. Meth. Appl. Mech. Eng. 336, 533–553 (2018)

    MathSciNet  MATH  Google Scholar 

  56. Chen, C.-Y., Meiburg, E.: Miscible porous media displacements in the quarter five-spot configuration: Part 1–The homogeneous case. J. Fluid Mech. 371, 233–268 (1998)

    MathSciNet  MATH  Google Scholar 

  57. Petitjeans, P., Chen, C.Y., Meiburg, E., Maxworthy, T.: Miscible quarter five-spot displacements in a Hele-Shaw cell and the role of flow-induced dispersion. Phys. Fluids 11, 1705–1716 (1999)

    MathSciNet  MATH  Google Scholar 

  58. Kim, M.-Y., Park, E.-J., Thomas, S.G., Wheeler, M.F.: A multiscale mortar mixed finite element method for slightly compressible flows in porous media. J. Korean Math. Soc. 44, 1103–1119 (2007)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Project Numbers 14304217 and 14302018), CUHK Faculty of Science Direct Grant 2019-20 and NSFC/RGC Joint Research Scheme (Project Number HKUST620/15). The research of Eun-Jae Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (NRF-2015R1A5A1009350 and NRF-2019R1A2C2090021).

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR, China

    Lina Zhao

  2. Hong Kong Centre for Cerebro-Cardiovascular Health Engineering, Hong Kong SAR, China

    Dohyun Kim

  3. School of Mathematics and Computing, Computational Science and Engineering, Yonsei University, Seoul, 03722, Korea

    Eun-Jae Park

  4. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China

    Eric Chung

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  1. Lina Zhao
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  2. Dohyun Kim
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  3. Eun-Jae Park
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  4. Eric Chung
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Correspondence to Eun-Jae Park.

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Zhao, L., Kim, D., Park, EJ. et al. Staggered DG Method with Small Edges for Darcy Flows in Fractured Porous Media. J Sci Comput 90, 83 (2022). https://doi.org/10.1007/s10915-022-01760-8

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