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A Div FOSLS Method Suitable for Quadrilateral RT and Hexahedral RTN \(H(\textrm{div})\)-elements

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Abstract

A new div FOSLS mixed finite element method is proposed and analyzed for the first-order system of the general second-order elliptic problems in terms of the scalar variable displacement and the vectorial variable flux. The main feature of the proposed method is to apply a local constant element or (bi,tri)linear element \(L^2\) projection to the div term, together with a mesh-dependent div term. The method is coercive and symmetric, allowing any combination of conforming approximations for both scalar and vectorial variables. More importantly, the method is suitable for general nonaffine quadrilateral Raviart–Thomas (RT) and nonaffine hexahedral Raviart–Thomas–Nédélec (RTN) \(H(\textrm{div})\)-elements. For nonaffine quadrilateral RT elements in two dimensions, the proposed method provides optimal approximations for the scalar and vectorial variables for the combination \(Q_m-\textbf{RT}_\ell \) for all \(m\ge 1, \ell \ge s\) where \(s=0\) corresponds to the constant element \(L^2\) projection and \(s=1\) corresponds to the bilinear element \(L^2\) projection; for nonaffine hexahedral RTN elements in three dimensions, it provides optimal approximations for the scalar variable for the combination \(Q_m-\textbf{RTN}_m\) for all \(m\ge 1\) while suboptimal approximations with order m for the vectorial variable. Numerical results are given to confirm the performance and the theoretical results of the new method.

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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. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Ambartsumyan, I., Khattatov, E., Lee, J., Yotov, I.: Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra, M3AS. Math. Model Method Appl. Sci. 29, 1037–1077 (2019)

    Article  MATH  Google Scholar 

  3. Arbogast, T., Correa, M.R.: Two families of H(div) mixed finite elements on quadrilaterals of minimal dimension. SIAM J. Numer. Anal. 54, 3332–3356 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arbogast, T., Tao, Z.: Construction of H(div)-conforming mixed finite elements on cuboidal hexahedra. Numer. Math. 142, 1–32 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Babus̆ka, I.: Error bound for finite element method. Numer. Math. 16, 322–333 (1971)

    Article  MathSciNet  Google Scholar 

  7. Bendali, A.: Composite mixed finite elements on plane quadrilaterals. Numer. Methods Part. Differ. Equ. 9, 551–559 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bernardi, C., Girault, V.: A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35, 1893–1916 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bramble, J.H., Lazarov, R.D., Pasciak, J.E.: A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comput. 66, 935–955 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  11. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  12. Cai, Z.Q., Ku, J.: The L2 norm error estimates for the Div least-squares method. SIAM J. Numer. Anal. 44, 1721–1734 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cai, Z.Q., Ku, J.: Optimal error estimates for the div least-squares method with data \(f\in L^2\) and application to nonlinear problems. SIAM J. Numer. Anal. 47, 4098–4111 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cai, Z.Q., Lazarov, R.D., Manteuffel, T.A., McCormick, S.F.: First-order system least-squares for second-order partial differential equations: part I. SIAM J. Numer. Anal. 31, 1785–1799 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cai, Z.Q., Manteuffel, T.A., McCormick, S.F.: First-order system least-squares for second-order partial differential equations: part II. SIAM J. Numer. Anal. 34, 425–454 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cai, Z.Q., Falgout, R., Zhang, S.: Div first-order system LL* (FOSLL*) for second-order elliptic partial differential equations. SIAM J. Numer. Anal. 53, 405–420 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, H., Fu, G., Li, J., Qiu, W.: First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems. Comput. Math. Appl. 68, 1635–1652 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, Vol. II, Finite Element Methods (part 1). North-Holland, Amsterdam (1991)

    Google Scholar 

  19. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Numer. Anal. 9, 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  20. Cockburn, B., Fu, G.: Superconvergence by M-decompositions. Part III: construction of three-dimensional finite elements. ESAIM: Math. Model. Numer. Anal., 51, 365–398 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Duan, H.Y., Liang, G.P.: Nonconforming elements in least-squares mixed finite element methods. Math. Comput. 73, 1–18 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Duan, H.Y., Liang, G.P.: Normal and tangential continuous elements for least-squares mixed finite element methods. Numer. Methods Part. Differ. Equ. 20, 609–623 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Duan, H.Y., Qiu, F.J.: A new stabilized finite element method for advection-diffusion-reaction equations. Numer. Methods Part. Differ. Equ. 32, 616–645 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Duan, H.Y., Lin, P., Saikrishnan, P., Tan, Roger C. E.: \(L^2\)-projected least-squares finite element methods for the Stokes equations. SIAM J. Numer. Anal. 44, 732–752 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Duan, H.Y., Lin, P., Saikrishnan, P., Tan, R.C.E.: A least squares finite element method for the magnetostatic problem in a multiply-connected Lipschitz domain. SIAM J. Numer. Anal. 45, 2537–2563 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Falk, R.S., Gatto, P., Monk, P.: Hexahedral H(div) and H(curl) finite elements. ESAIM Math. Model. Numer. Anal. 45, 115–143 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  28. Jiang, B.N.: The Least-Squares Finite Element Method. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  29. Kwak, D.Y., Pyo, H.C.: Mixed finite element methods for general quadrilateral grids. Appl. Math. Comput. 217, 6556–6565 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Li, K.T., Huang, A.X., Huang, Q.H.: Finite Element Method and Applications (in Chinese), Revised Edition, Xi’an Jiaotong University Press. Xi’an, China (1992)

  31. Nédélec, J.C.: Mixed finite elements in \(R^3\). Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nédélec, J.C.: A new family of mixed finite elements in \(R^3\). Numer. Math. 50, 57–81 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pehlivanov, A.I., Carey, G.F., Lazarov, R.D.: Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal. 31, 1368–1377 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pehlivanov, A.I., Carey, G.F., Vassilevski, P.S.: Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates. Numer. Math., 72, 501–522 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Raviart, R.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.), Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, Springer-Verlag, New York, pp. 292-315 (1977)

  36. Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 2, pp. 523–639. North-Holland, Amsterdam (1991)

    Google Scholar 

  37. Sboui, A., Jaffré, J., Roberts, J.: A composite mixed finite element for hexahedral grids. SIAM J. Sci. Comput. 31, 2623–2645 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Thomas, J.M.: Sur l’analyse numerique des methodes d’elements nis hybrides et mixtes, Ph.D. Thesis, Université Pierre et Marie Curie, Paris, France (1977)

  39. Wang, J., Mathew, T.: Mixed finite element methods over quadrilaterals. In: Dimov, J.T., Sendov, B., Vassilevski, P. (eds.) Proceedings of 3rd International Conference on Advances in Numerical Methods and Applications, pp. 203–214. World Scientific, River Edge, NJ (1994)

  40. Yang, D.: Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems. Math. Comput. 69, 929–963 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by National Natural Science Foundation of China (11971366, 11571266 and 11661161017 ) and by the Hubei Key Laboratory of Computational Science, Wuhan University, China (2019CFA007).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Huoyuan Duan, Can Wang & Zhijie Du

  2. School of Artificial Intelligence, Jianghan University, Wuhan, 430056, China

    Zhijie Du

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  1. Huoyuan Duan
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  2. Can Wang
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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Huoyuan Duan.

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Duan, H., Wang, C. & Du, Z. A Div FOSLS Method Suitable for Quadrilateral RT and Hexahedral RTN \(H(\textrm{div})\)-elements. J Sci Comput 93, 85 (2022). https://doi.org/10.1007/s10915-022-02043-y

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