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Error Estimates of EDG-HDG Methods for the Stokes Equations with Dirac Measures

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Abstract

In this paper, we analyze the hybridized, embedded-hybridized and embedded discontinuous Galerkin methods for the Stokes equations with Dirac measures. The velocity, the velocity traces and the pressure traces are approximated by polynomials of degree \(k\ge 1\), and the pressure is discretized by polynomials of degree \(k-1\). An attractive property, named divergence-free, is satisfied by the discrete velocity field. Moreover, the discrete velocity fields derived by hybridized and embedded-hybridized discontinuous Galerkin methods are H(div)-conforming. Using duality argument and Oswald interpolation, a priori and a posteriori error estimates are obtained for the velocity in \(L^2\)-norm. In addition, a posteriori error estimates for the velocity in \(W^{1,q}\)-seminorm and the pressure in \(L^q\)-norm are also derived. Finally, numerical examples are provided to validate the theoretical analysis and show the performance of the obtained a posteriori error estimators.

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References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, Cambridge (1975)

    MATH  Google Scholar 

  2. Araya, R., Behrens, E., Rodríguez, R.: A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math. 105, 193–216 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Agnelli, J.P., Garau, E.M., Morin, P.: A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces. ESAIM: Math. Model. Numer. Anal. 48, 1557–1581 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Apel, T., Benedix, O., Sirch, D., Vexler, B.: A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 49, 992–1005 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Allendes, A., Otárola, E., Rankin, R., Salgado, A.J.: Adaptive finite element methods for an optimal control problem involving Dirac measures. Numer. Math. 137, 159–197 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Allendes, A., Otárola, E., Salgado, A.J.: A posteriori error estimates for the Stokes problem with singular sources. Comput. Methods Appl. Mech. Engrg. 345, 1007–1032 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Allendes, A., Fuica, F., Otárola, E., Quero, D.: An adaptive FEM for the Pointwise tracking optimal control problem of the Stokes equations. SIAM J. Sci. Comput. 41, A2967–A2998 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Babuška, I.: Error bounds for the finite element method. Numer. Math. 16, 322–333 (1971)

    MathSciNet  MATH  Google Scholar 

  9. Brown, R.M., Shen, Z.: Estimates for the Stokes operator in Lipschitz domains. Indiana Univ. Math. J. 44, 1183–1206 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (2008)

    MATH  Google Scholar 

  11. Bernardi, C., Canuto, C., Maday, Y.: Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25, 1237–1271 (1988)

    MathSciNet  MATH  Google Scholar 

  12. Brett, C., Dedner, A., Elliott, C.: Optimal control of elliptic PDEs at points. IMA J. Numer. Anal. 36, 1015–1050 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Bertoluzza, S., Decoene, A., Lacouture, L., Martin, S.: Local error analysis for the Stokes equations with a punctual source term. Numer. Math. 140, 677–701 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47, 1092–1125 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.-J.: Analysis of HDG methods for Stokes flow. Math. Comput. 80, 723–760 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Cockburn, B., Sayas, F.-J.: Divergence-conforming HDG methods for Stokes flows. Math. Comput. 83, 1571–1598 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Cockburn, B., Guzmán, J., Soon, S.-C., Stolarski, H.K.: An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems. SIAM J. Numer. Anal. 47, 2686–2707 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Chen, H., Li, J., Qiu, W.: Robust a posteriori error estimates for HDG method for convection diffusion equations. IMA J. Numer. Anal. 36, 437–462 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Chen, H., Qiu, W., Shi, K.: A priori and computable a posteriori error estimates for an HDG method for the coercive Maxwell equations. Comput. Methods Appl. Mech. Engrg. 333, 287–310 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Fulford, G., Blake, J.: Muco-ciliary transport in the lung. J. Theor. Biol. 121, 381–402 (1986)

    Google Scholar 

  22. Fuica, F., Lepe, F., Otárola, E., Quero, D.: A posteriori error estimates in \(W^{1,p}\times L^{p}\) spaces for the Stokes system with Dirac measures (2019) arXiv:1912.08325

  23. Fuica, F., Otárola, E., Quero, D.: Error estimates for optimal control problems invovling the Stokes system and Dirac measures. Appl. Math. Optim. 84, 1717–1750 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Fu, G., Qiu, W., Zhang, W.: An analysis of HDG methods for convection-dominated diffusion problems. ESAIM: M2AN 49, 225–256 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations: steady-state problems. Springer, New York (2011)

    MATH  Google Scholar 

  26. Houston, P., Wihler, T.P.: Discontinuous Galerkin methods for problems with Dirac delta source, ESAIM: Math. Model. Numer. Anal. 46, 1467–1483 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Ignacio, O.: Optimal a priori error estimates in weighted Sobolev spaces for the Poisson problem with singular sources, ESAIM: Math. Model. Numer. Anal. 55, s879–s907 (2021)

    Google Scholar 

  28. Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Lacouture, L.: A numerical method to solve the Stokes problem with punctual force in source term. Comptes Rendus Mcanique 343, 187–191 (2015)

    Google Scholar 

  30. Labeur, R.J., Wells, G.N.: A Galerkin interface stabilization method for the advection-diffusion and incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 196, 4985–5000 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Labeur, R.J., Wells, G.N.: Energy stable and momentum conserving hybrid finite element method for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 32, A889–A913 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Lepe, F., Otárola, E., Quero, D.: Error estimates for FEM discretizations of the Navier-Stokes equations with Dirac measures. J. Sci. Comput. 87, 97 (2021)

    MathSciNet  MATH  Google Scholar 

  33. Leng, H., Chen, Y.: A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source, ESAIM: Math. Model. Numer. Anal. 56, 385–406 (2022)

    MathSciNet  MATH  Google Scholar 

  34. Maz’ya, V.G., Rossmann, J.: Lp estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains. Math. Nachr. 7, 751–793 (2007)

    MATH  Google Scholar 

  35. Mitrea, M., Wright, M.: Bouneary value problems for the Stokes system in arbitrary Lipschitz domains, Astérisque, viii+241 (2012)

  36. Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridization discontinuous Galerkin method for Stokes flow, Comput. Methods, Appl. Mech. Engrg. 199, 582–597 (2010)

    MATH  Google Scholar 

  37. Nochetto, R.H., Otárola, E., Salgado, A.J.: Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132, 85–130 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87, 69–93 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Rhebergen, S., Wells, G.N.: Analysis of a hybridized/interface stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 55, 1982–2003 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Rhebergen, S., Wells, G.N.: An embedded-hybridized discontinuous Galerkin finite element method for the Stokes equations. Comput. Methods Appl. Mech. Engrg. 358, 112619 (2020)

    MathSciNet  MATH  Google Scholar 

  41. Svensson, E. D., Larsson, S.: Pointwise a posteriori error estimates for the Stokes equations in polyhedral domains, preprint (2006)

  42. Scott, R.: Finite element convergence for singular data. Numer. Math. 21, 317–327 (1973)

    MathSciNet  MATH  Google Scholar 

  43. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley and Teubner, New York (1996)

    MATH  Google Scholar 

  44. Verfürth, R.: A posteriori error estimators for convection-diffusion equations. Numer. Math. 80, 641–663 (1998)

    MathSciNet  MATH  Google Scholar 

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  1. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510000, Guangdong, China

    Haitao Leng

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The work was supported by the NSF of China (Grant No. 12001209).

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Leng, H. Error Estimates of EDG-HDG Methods for the Stokes Equations with Dirac Measures. J Sci Comput 94, 66 (2023). https://doi.org/10.1007/s10915-023-02116-6

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