Skip to main content
SpringerLink
Log in

Continuous Galerkin and Enriched Galerkin Methods with Arbitrary Order Discontinuous Trial Functions for the Elliptic and Parabolic Problems with Jump Conditions

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, a new version of the enriched Galerkin (EG) method for elliptic and parabolic equations is presented and analyzed, which is capable of dealing with a jump condition along a submanifold \({\Gamma _{\text {LG}}}\). The jump condition is known as Henry’s law in a stationary diffusion process. Here, the novel EG finite element method is constructed by enriching the continuous Galerkin finite element space by not only piecewise constants but also with piecewise polynomials with an arbitrary order. In addition, we extend the proposed method to consider new versions of a continuous Galerkin (CG) and a discontinuous Galerkin (DG) finite element method. The presented uniform analyses for CG, DG, and EG account for a spatially and temporally varying diffusion tensor which is also allowed to have a jump at \({\Gamma _{\text {LG}}}\) and gives optimal convergence results. Several numerical experiments verify the presented analyses and illustrate the capability of the proposed methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 1150013-1–1150013-40 (2012)

  2. Arndt, D., Bangerth, W., Clevenger, T.C., Davydov, D., Fehling, M., Garcia-Sanchez, D., Harper, G., Heister, T., Heltai, L., Kronbichler, M., Kynch, R.M., Maier, M., Pelteret, J.-P., Turcksin, B., Wells, D.: The deal.II library, version 9.1. J. Numer. Math. (2019, accepted). https://doi.org/10.1515/jnma-2019-0064. https://dealii.org/deal91-preprint.pdf

  3. Becker, R., Burman, E., Hansbo, P., Larson, M.G.: A reduced P1-discontinuous Galerkin method. Chalmers Finite Element Center Preprint 2003-13 (2003)

  4. Chabaud, B., Cockburn, B.: Uniform-in-time superconver-gence of HDG methods for the heat equation. Math. Comput. 81, 107–129 (2012)

    Article  Google Scholar 

  5. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numerische Mathematik 79, 175–202 (1998). https://doi.org/10.1007/s002110050336

    Article  MathSciNet  MATH  Google Scholar 

  6. Choo, J., Lee, S.: Enriched Galerkin finite elements for coupled poromechanics with local mass conservation. Comput. Methods Appl. Mech. Eng. 341, 311–332 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, Mathmatiques et Applications. Springer, Heidelberg (2012)

    Google Scholar 

  9. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences. Springer, New York (2004)

    Book  Google Scholar 

  10. Huang, J., Zou, J.: Some new a priori estimates for second-order elliptic and parabolic interface problems. J. Differ. Equ. 184, 570–586 (2002)

    Article  MathSciNet  Google Scholar 

  11. Jäger, W., Mikelić, A., Neuss-Radu, M.: Analysis of differential equations modelling the reactive flow through a deformable system of cells. Arch. Ration. Mech. Anal. 192, 331–374 (2009). https://doi.org/10.1007/s00205-008-0118-4

  12. Kadeethum, T., Nick, H., Lee, S.: Comparison of two-and three-field formulation discretizations for flow and solid deformation in heterogeneous porous media. In: 20th Annual Conference of the International Association for Mathematical Geosciences (2019)

  13. Kadeethum, T., Nick, H., Lee, S., Richardson, C., Salimzadeh, S., Ballarin, F.: A novel enriched Galerkin method for modelling coupled flow and mechanical deformation in heterogeneous porous media. In: 53rd US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association, New York, NY, USA (2019)

  14. Kadeethum, T., Nick, H.M., Lee, S., Ballarin, F.: Flow in porous media with low dimensional fractures by employing enriched Galerkin method. Adv. Water Resour. (2020). https://doi.org/10.1016/j.advwatres.2020.103620

    Article  Google Scholar 

  15. Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics. Springer, New York (2003). https://doi.org/10.1007/b97419

    Book  MATH  Google Scholar 

  16. Kuzmin, D., Hajduk, H., Rupp, A.: Locally bound-preserving enriched Galerkin methods for the linear advection equation. Comput. Fluids 205, 15 (2020). https://doi.org/10.1016/j.compfluid.2020.104525

    Article  MathSciNet  MATH  Google Scholar 

  17. Lee, S., Choi, W.: Optimal error estimate of elliptic problems with Dirac sources for discontinuous and enriched Galerkin methods. Appl. Numer. Math. (2019). https://doi.org/10.1016/j.apnum.2019.09.010

    Article  MATH  Google Scholar 

  18. Lee, S., Lee, Y.-J., Wheeler, M.F.: A locally conservative enriched Galerkin approximation and efficient solver for elliptic and parabolic problems. SIAM J. Sci. Comput. 38, A1404–A1429 (2016)

    Article  MathSciNet  Google Scholar 

  19. Lee, S., Mikelic, A., Wheeler, M.F., Wick, T.: Phase-field modeling of two phase fluid filled fractures in a poroelastic medium. Multiscale Model. Simul. 16, 1542–1580 (2018)

    Article  MathSciNet  Google Scholar 

  20. Lee, S., Wheeler, M.F.: Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization. J. Comput. Phys. 331, 19–37 (2017)

    Article  MathSciNet  Google Scholar 

  21. Lee, S., Wheeler, M.F.: Enriched Galerkin methods for two-phase flow in porous media with capillary pressure. J. Computat. Phys. 367, 65–86 (2018)

    Article  MathSciNet  Google Scholar 

  22. Lehrenfeld, C., Reusken, A.: High Order Unfitted Finite ElementMethods for Interface Problems and PDEs on Surfaces, pp. 33–63. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56602-3_2

    Book  MATH  Google Scholar 

  23. Muntean, A., Böhm, M.: A moving-boundary problem for concrete carbonation: global existence and uniqueness of weak solutions. J. Math. Anal. Appl. 350, 234–251 (2009). https://doi.org/10.1016/j.jmaa.2008.09.044

    Article  MathSciNet  MATH  Google Scholar 

  24. Rupp, A.: Simulating Structure Formation in Soils Across Scales Using Discontinuous Galerkin Methods, Mathematik, Shaker Verlag GmbH. Düren 07 (2019). https://doi.org/10.2370/9783844068016

  25. Rupp, A., Knabner, P., Dawson, C.: A local discontinuous Galerkin scheme for Darcy flow with internal jumps. Comput. Geosci. 22, 1149–1159 (2018). https://doi.org/10.1007/s10596-018-9743-7

    Article  MathSciNet  MATH  Google Scholar 

  26. Rupp, A., Totsche, K.U., Prechtel, A., Ray, N.: Discrete-continuum multiphase model for structure formation in soils including electrostatic effects. Front. Environ. Sci. 6, 13 (2018). https://doi.org/10.3389/fenvs.2018.00096

    Article  Google Scholar 

  27. Sander, R.: Compilation of Henry’s law constants (version 4.0) for water as solvent. Atmos. Chem. Phys. 15, 4399–4981 (2015). https://doi.org/10.5194/acp-15-4399-2015

    Article  Google Scholar 

  28. Sochala, P., Ern, A., Piperno, S.: Mass conservative BDF-discontinous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows. Comput. Methods Appl. Mech. Eng. 198, 2122–2136 (2009). https://doi.org/10.1016/j.cma.2009.02.024

  29. Sun, S., Liu, J.: A locally conservative finite element method based on piecewise constant enrichment of the continuous Galerkin method. SIAM J. Sci. Comput. 31, 2528–2548 (2009)

    Article  MathSciNet  Google Scholar 

  30. Vamaraju, J., Sen, M., Basabe, J.D., Wheeler, M.: A comparison of continuous, discontinuous, and enriched Galerkin finite-element methods for elastic wave-propagation simulation. pp. 4063–4067 (2017). https://doi.org/10.1190/segam2017-17658225.1

Download references

Acknowledgements

A. Rupp acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster). S. Lee is supported by the National Science Foundation under Grant No. (NSF DMS-1913016).

Author information

Authors and Affiliations

  1. Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Mathematikon, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Andreas Rupp

  2. Department of Mathematics, Florida State University, 208 Love Building 1017 Academic Way, Tallahassee, FL, 32306-4510, USA

    Sanghyun Lee

Authors
  1. Andreas Rupp
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanghyun Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanghyun Lee.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Submitted to the editors DATE.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rupp, A., Lee, S. Continuous Galerkin and Enriched Galerkin Methods with Arbitrary Order Discontinuous Trial Functions for the Elliptic and Parabolic Problems with Jump Conditions. J Sci Comput 84, 9 (2020). https://doi.org/10.1007/s10915-020-01255-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01255-4

Keywords

Mathematics Subject Classification

Search

Navigation