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Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

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Abstract

This paper is devoted to investigating the optimal convergence order of a weak Galerkin finite element approximation to a second-order parabolic equation whose solution has lower regularity. In many applications, the solution of a second-order parabolic equation has only \(\varvec{H}^{\varvec{1+s}}\) smoothness with \(\varvec{0<s<1}\), and the numerical experiments show that the weak Galerkin approximate solution exhibits an optimal convergence order of \(\varvec{1+s}\). However, the standard numerical analysis for weak Galerkin finite element method always requires that the exact solution should have at least \(\varvec{H}^{\varvec{2}}\) smoothness. Our work fills the gap in the error analysis of weak Galerkin finite element method under lower regularity condition, where we prove the convergence order is of \(\varvec{1+s}\). The main strategy of analysis is to introduce an \(\varvec{H}^{\varvec{2}}\)-regular finite element approximation to discretize the spatial variables in variational equation, and then we analyze the error between this semi-discretized solution and the full discretized weak Galerkin solution. Finally, we present some numerical experiments to validate the theoretical analysis.

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References

  1. Mu, L., Wang, J., Wang, Y., Ye, X.: A computational study of the weak Galerkin method for second-order elliptic equations. Numer. Algorithms 63(4), 753–777 (2013)

    MathSciNet  Google Scholar 

  2. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)

    MathSciNet  Google Scholar 

  3. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Math., vol. 606, pp. 292–315. Springer, (1977)

  4. Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)

    MathSciNet  Google Scholar 

  5. Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comput. Appl. Math. 285, 45–58 (2015)

    MathSciNet  Google Scholar 

  6. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. Int. J. Numer. Anal. Model. 12(1), 31–53 (2015)

    MathSciNet  Google Scholar 

  7. Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comp. 83(289), 2101–2126 (2014)

    MathSciNet  Google Scholar 

  8. Zhou, C., Zou, Y., Chai, S., Zhang, F.: Mixed weak Galerkin method for heat equation with random initial condition. Math. Probl. Eng. 8796345–11 (2020)

  9. Zhou, C., Zou, Y., Chai, S., Zhang, Q., Zhu, H.: Weak Galerkin mixed finite element method for heat equation. Appl. Numer. Math. 123, 180–199 (2018)

    MathSciNet  Google Scholar 

  10. Peng, H., Zhai, Q.: Weak Galerkin method for the Stokes equations with damping. Discrete Contin. Dyn. Syst. Ser. B 27(4), 1853–1875 (2022)

    MathSciNet  Google Scholar 

  11. Tian, T., Zhai, Q., Zhang, R.: A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. 329, 268–279 (2018)

    MathSciNet  Google Scholar 

  12. Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math. 42(1), 155–174 (2016)

    MathSciNet  Google Scholar 

  13. Wang, R., Wang, X., Zhai, Q., Zhang, R.: A weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. 302, 171–185 (2016)

    MathSciNet  Google Scholar 

  14. Zhai, Q., Zhang, R., Wang, X.: A hybridized weak Galerkin finite element scheme for the Stokes equations. Sci. China Math. 58(11), 2455–2472 (2015)

    MathSciNet  Google Scholar 

  15. Mu, L., Wang, J., Ye, X., Zhang, S.: A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput. 65(1), 363–386 (2015)

    MathSciNet  Google Scholar 

  16. Shields, J., Machorro, E.A.: Weak Galerkin methods for time-dependent Maxwell’s equations. Comput. Math. Appl. 74(9), 2106–2124 (2017)

    MathSciNet  Google Scholar 

  17. Mu, L., Wang, J., Ye, X.: A new weak Galerkin finite element method for the Helmholtz equation. IMA J. Numer. Anal. 35(3), 1228–1255 (2015)

    MathSciNet  Google Scholar 

  18. Mu, L., Wang, J., Ye, X., Zhao, S.: A numerical study on the weak Galerkin method for the Helmholtz equation. Commun. Comput. Phys. 15(5), 1461–1479 (2014)

    MathSciNet  Google Scholar 

  19. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer. Methods Partial Differential Equations 30(3), 1003–1029 (2014)

    MathSciNet  Google Scholar 

  20. Zhang, R., Zhai, Q.: A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J. Sci. Comput. 64(2), 559–585 (2015)

    MathSciNet  Google Scholar 

  21. Chai, S., Wang, Y., Zhao, W., Zou, Y.: A \(C^0\) weak Galerkin method for linear Cahn-Hilliard-Cook equation with random initial condition. Appl. Math. Comput. 414, 126659–11 (2022)

    Google Scholar 

  22. Wang, J., Zhai, Q., Zhang, R., Zhang, S.: A weak Galerkin finite element scheme for the Cahn-Hilliard equation. Math. Comp. 88(315), 211–235 (2019)

    MathSciNet  Google Scholar 

  23. Zhu, H., Zou, Y., Chai, S., Zhou, C.: Numerical approximation to a stochastic parabolic PDE with weak Galerkin method. Numer. Math. Theory Methods Appl. 11(3), 604–617 (2018)

    MathSciNet  Google Scholar 

  24. Zhu, H., Zou, Y., Chai, S., Zhou, C.: A weak Galerkin method with RT elements for a stochastic parabolic differential equation. East Asian J. Appl. Math. 9(4), 818–830 (2019)

    MathSciNet  Google Scholar 

  25. Sun, L., Feng, Y., Liu, Y., Zhang, R.: The modified weak Galerkin finite element method for solving Brinkman equations. J. Math. Res. Appl. 39(6), 657–676 (2019)

    MathSciNet  Google Scholar 

  26. Wang, X., Zhai, Q., Zhang, R.: The weak Galerkin method for solving the incompressible Brinkman flow. J. Comput. Appl. Math. 307, 13–24 (2016)

    MathSciNet  Google Scholar 

  27. Peng, H., Wang, R., Wang, X., Zou, Y.: Weak Galerkin finite element method for linear elasticity interface problems. Appl. Math. Comput. 439, 127589–12 (2023)

    MathSciNet  Google Scholar 

  28. Zhao, L., Wang, R., Zou, Y.: A hybridized weak Galerkin finite element scheme for linear elasticity problem. J. Comput. Appl. Math. 425, 115024–15 (2023)

    MathSciNet  Google Scholar 

  29. Zhai, Q., Hu, X., Zhang, R.: The shifted-inverse power weak Galerkin method for eigenvalue problems. J. Comput. Math. 38(4), 606–623 (2020)

    MathSciNet  Google Scholar 

  30. Zhai, Q., Xie, H., Zhang, R., Zhang, Z.: The weak Galerkin method for elliptic eigenvalue problems. Commun. Comput. Phys. 26(1), 160–191 (2019)

    MathSciNet  Google Scholar 

  31. Zhai, Q., Zhang, R.: Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes. Discrete Contin. Dyn. Syst. Ser. B 24(1), 403–413 (2019)

    MathSciNet  Google Scholar 

  32. Guo, N., Chen, Y., Deng, X.: A lifting method for Krause’s consensus model. Commun. Math. Res. 38(1), 52–61 (2022)

    MathSciNet  Google Scholar 

  33. Thomée, V.: Finite difference methods for linear parabolic equations. Handbook of numerical analysis, vol. I, pp. 5–196. North-Holland, Amsterdam (1990)

  34. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal. 32(6), 1729–1749 (1995)

  35. Thomée, V.: Galerkin finite element methods for parabolic problems, 2nd edn. Springer Series in Computational Mathematics, vol. 25, p. 370. Springer, (2006)

  36. Chatzipantelidis, P., Lazarov, R.D., Thomée, V.: Error estimates for a finite volume element method for parabolic equations in convex polygonal domains. Numer. Methods Partial Differential Equations 20(5), 650–674 (2004)

    MathSciNet  Google Scholar 

  37. Yang, M., Liu, J.: A quadratic finite volume element method for parabolic problems on quadrilateral meshes. IMA J. Numer. Anal. 31(3), 1038–1061 (2011)

    MathSciNet  Google Scholar 

  38. Larsson, S., Thomée, V., Wahlbin, L.B.: Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comp. 67(221), 45–71 (1998)

    MathSciNet  Google Scholar 

  39. Makridakis, C.G., Babuška, I.: On the stability of the discontinuous Galerkin method for the heat equation. SIAM J. Numer. Anal. 34(1), 389–401 (1997)

    MathSciNet  Google Scholar 

  40. Dutt, P., Biswas, P., Ghorai, S.: Spectral element methods for parabolic problems. J. Comput. Appl. Math. 203(2), 461–486 (2007)

    MathSciNet  Google Scholar 

  41. Suhov, A.Y.: A spectral method for the time evolution in parabolic problems. J. Sci. Comput. 29(2), 201–217 (2006)

    MathSciNet  Google Scholar 

  42. Chai, S., Zou, Y., Zhao, W.: A weak Galerkin method for \(C^0\) element for fourth order linear parabolic equation. Adv. Appl. Math. Mech. 11(2), 467–485 (2019)

    MathSciNet  Google Scholar 

  43. Chai, S., Zou, Y., Zhou, C., Zhao, W.: Weak Galerkin finite element methods for a fourth order parabolic equation. Numer. Methods Partial Differential Equations 35(5), 1745–1755 (2019)

    MathSciNet  Google Scholar 

  44. Li, Q.H., Wang, J.: Weak Galerkin finite element methods for parabolic equations. Numer. Methods Partial Differential Equations 29(6), 2004–2024 (2013)

    MathSciNet  Google Scholar 

  45. Zhang, H., Zou, Y., Chai, S., Yue, H.: Weak Galerkin method with \((r, r-1, r-1)\)-order finite elements for second order parabolic equations. Appl. Math. Comput. 275, 24–40 (2016)

    MathSciNet  Google Scholar 

  46. Zhang, H., Zou, Y., Xu, Y., Zhai, Q., Yue, H.: Weak Galerkin finite element method for second order parabolic equations. Int. J. Numer. Anal. Model. 13(4), 525–544 (2016)

    MathSciNet  Google Scholar 

  47. Wu, Z., Yin, J., Wang, C.: Elliptic & Parabolic equations, p. 408. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2006)

  48. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods, 3rd edn. Texts in Applied Mathematics, vol. 15, p. 397. Springer, (2008)

  49. Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19, p. 662. American Mathematical Society, Providence, RI, (1998)

  50. Wang, Y., Zou, Y., Chai, S.: \((1+s)\)-order convergence analysis of weak Galerkin finite element methods for second order elliptic equations. Adv. Appl. Math. Mech. 13(3), 554–568 (2021)

    MathSciNet  Google Scholar 

  51. Charrier, J., Scheichl, R., Teckentrup, A.L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322–352 (2013)

    MathSciNet  Google Scholar 

  52. Hackbusch, W.: Elliptic differential equations: theory and numerical treatment, 2nd edn. Springer Series in Computational Mathematics, vol. 18, p. 455. Springer, (2017)

  53. Málek, J., Nečas, J., Rokyta, M., Ružička, M.: Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13, p. 317. Chapman & Hall, London, (1996)

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Acknowledgements

The authors are grateful to the reviewers for their constructive suggestions and comments to improve the paper.

Funding

NSFC (12171199, 11971198, 12371422), the National Key R &D Program (2020YFA0714101, 2020YFA0713601), Jilin Provincial Department S &T (20210201015GX, 20210201078GX), Jilin Provincial Foundation (YDZJ202201ZYTS535, JJKH20220151KJ).

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Xuan Liu, Yongkui Zou & Shimin Chai

  2. College of Applied Mathematics, Jilin University of Finance and Economics, Changchun, 130117, China

    Huimin Wang

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  1. Xuan Liu
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Y. Zou and S. Chai developed the idea for the study; X. Liu conducted the analyses and wrote the initial draft of the paper; all authors discussed the results and revised the manuscript.

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Correspondence to Shimin Chai.

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Liu, X., Zou, Y., Chai, S. et al. Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity. Numer Algor 97, 1323–1339 (2024). https://doi.org/10.1007/s11075-024-01751-w

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