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An Optimal Multigrid Algorithm for the Combining \(P_1\)-\(Q_1\) Finite Element Approximations of Interface Problems Based on Local Anisotropic Fitting Meshes

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Abstract

A new finite element method is proposed for second order elliptic interface problems based on a local anisotropic fitting mixed mesh. The local anisotropic fitting mixed mesh is generated from an interface-unfitted mesh by simply connecting the intersected points of the interface and the underlying mesh successively. Optimal approximation capabilities on anisotropic elements are proved, the convergence rates are linear and quadratic in \(H^1\) and \(L^2\) norms, respectively. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a new multigrid method is presented to handle this issue. The convergence rate of the multigrid method is shown to be optimal with respect to both the coefficient jump ratio and mesh size. Numerical experiments are presented to demonstrate the theoretical results.

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References

  1. Acosta, G., Duran, R.G.: Error estimates for \(Q_1\) isoparametric elements satisfying a weak angle condition. SIAM J. Num. Anal. 38, 1073–1088 (2000)

    Article  Google Scholar 

  2. Adams, R. A., Fournier, J. J.: Sobolev Spaces. Academic press, (2003)

  3. Adjerid, S., Chaabane, N., Lin, T.: An immersed discontinuous finite element method for Stokes interface problems. Comput. Methods Appl. Mech. Eng. 293, 170–190 (2015)

    Article  MathSciNet  Google Scholar 

  4. Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970)

    Article  MathSciNet  Google Scholar 

  5. Babuška, I., Aziz, A.K.: On the angle condition in the finite element method. SIAM J. Num. Anal. 13, 214–226 (1976)

    Article  MathSciNet  Google Scholar 

  6. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Num. Methods Eng. 45, 601–620 (1999)

    Article  Google Scholar 

  7. Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6, 109–138 (1996)

    Article  MathSciNet  Google Scholar 

  8. Burman, E., Guzmán, J., Sánchez, M.A., Sarkis, M.: Robust flux error estimation of an unfitted Nitsche method for high-contrast interface problems. IMA Journal of Numerical Analysis (2016)

  9. Chen, L., Wei, H., Wen, M.: An interface-fitted mesh generator and virtual element methods for elliptic interface problems. J. Comput. Phys. 334, 327–348 (2017)

    Article  MathSciNet  Google Scholar 

  10. Chen, Z., Wu, Z., Xiao, Y.: An adaptive immersed finite element method with arbitrary Lagrangian-Eulerian scheme for parabolic equations in time variable domains. Int. J. Num. Analy. Model. 12, 567–591 (2015)

    MathSciNet  Google Scholar 

  11. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numerische Mathematik 79, 175–202 (1998)

    Article  MathSciNet  Google Scholar 

  12. Cumsille, P., Asenjoc, J., Conca, C.: A novel model for biofilm growth and its resolution by using the hybrid immersed interface-level set method. Comput. Math. Appl. 67, 34–51 (2014)

    Article  MathSciNet  Google Scholar 

  13. Guzmán, J., Sánchez, M., Sarkis, M.: On the accuracy of finite element approximations to a class of interface problems. Math. Comput. 85(301), 2071–2098 (2016)

    Article  MathSciNet  Google Scholar 

  14. Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191, 5537–5552 (2002)

    Article  MathSciNet  Google Scholar 

  15. Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193, 3523–3540 (2004)

    Article  MathSciNet  Google Scholar 

  16. Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Num. Math. 85, 90–114 (2014)

    Article  MathSciNet  Google Scholar 

  17. Hou, T., Li, Z., Osher, S., Zhao, H.: A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys. 134, 236–252 (1997)

    Article  MathSciNet  Google Scholar 

  18. Huang, J., Zou, J.: Some new a priori estimates for second order elliptic and parabolic interface problems. J. Diff. Equa. 184, 570–586 (2002)

    Article  MathSciNet  Google Scholar 

  19. Huang, J., Zou, J.: Uniform a priori estimates for elliptic and static Maxwell interface problems. Dis. Contin. Dynam. Syst.-Series B 7, 145–170 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Kergrene, K., Babuška, I., Banerjee, U.: Stable generalized finite element method and associated iterative schemes; application to interface problems. Comput. Methods Appl. Mech. Eng. 305, 1–36 (2016)

    Article  MathSciNet  Google Scholar 

  21. Li, Z.: The immersed interface method using a finite element formulation. Appl. Num. Math. 27, 253–267 (1998)

    Article  MathSciNet  Google Scholar 

  22. Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problems using the finite element formulation. Numerische Mathematik 96, 61–98 (2003)

    Article  MathSciNet  Google Scholar 

  23. Ma, Q., Cui, J., Li, Z., Wang, Z.: Second-order asymptotic algorithm for heat conduction problems of periodic composite materials in curvilinear coordinates. J. Comput. Appl. Math. 306, 85–115 (2016)

    Article  MathSciNet  Google Scholar 

  24. Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994)

    Article  Google Scholar 

  25. Xiao, Y., Xu, J., Wang, F.: High order eXtended finite element methods for interface problems. Comput. Methods Appl. Mech. Eng. 364, 1–21 (2020)

    Article  MathSciNet  Google Scholar 

  26. Xu, J.: Estimate of the convergence rate of finite element solutions to elliptic equations of second order with discontinuous coefficients. Natl. Sci. J. Xiangtan Univ. (in Chinese) 1, 84–88 (1982)

    MATH  Google Scholar 

  27. Xu, J., Zhang, S.: Optimal finite element methods for interface problems. Domain Decomposition Methods in Science and Engineering XXII, pages 77–91, (2016)

  28. Xu, J., Zhu, Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Model. Methods Appl. Sci. 18, 77–105 (2008)

    Article  MathSciNet  Google Scholar 

  29. Xu, J., Zikatanov, L.: The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15, 573–597 (2002)

    Article  MathSciNet  Google Scholar 

  30. Zhang, Y., Nguyen, D., Du, K., Xu, J., Zhao, S.: Time-domain numerical solutions of Maxwell interface problems with discontinuous electromagnetic waves. Adv. Appl. Math. Mech. 8, 353–385 (2016)

    Article  MathSciNet  Google Scholar 

  31. Zi, G., Belytschko, T.: New crack-tip elements for XFEM and applications to cohesive cracks. Int. J. Num. Methods Eng. 57, 2221–2240 (2003)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, China

    Jun Hu & Hua Wang

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  1. Jun Hu
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  2. Hua Wang
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Correspondence to Hua Wang.

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In this research, Jun Hu was supported by NSFC projects 11625101 and 11421101; Hua Wang was supported by China Postdoctoral Science Foundation Grand 2019M660277 and Jiangsu Key Lab for NSLSCS Grant 201906.

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Hu, J., Wang, H. An Optimal Multigrid Algorithm for the Combining \(P_1\)-\(Q_1\) Finite Element Approximations of Interface Problems Based on Local Anisotropic Fitting Meshes. J Sci Comput 88, 16 (2021). https://doi.org/10.1007/s10915-021-01536-6

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  • DOI: https://doi.org/10.1007/s10915-021-01536-6

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