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Accurate gradient computations at interfaces using finite element methods Cover

Accurate gradient computations at interfaces using finite element methods

Open Access
|Sep 2017

References

  1. Adams, R. and Fournier, J. (2003). Sobolev Spaces. Second Edition, Academic Press, Cambridge, MA.
  2. An, N. and Chen, H. (2014). A partially penalty immersed interface finite element method for anisotropic elliptic interface problems, Numerical Methods for Partial Differential Equations30(6): 1984–2028.10.1002/num.21886
  3. Anitescu, C. (2017). Open source 3D Matlab isogeometric analysis code, https://sourceforge.net/u/cmechanicsos/profile/.
  4. Babuška, I. (1970). The finite element method for elliptic equations with discontinuous coefficients, Computing5(3): 207–213.10.1007/BF02248021
  5. Bramble, J. and King, J. (1996). A finite element method for interface problems in domains with smooth boundaries and interfaces, Advances in Computational Mathematics6(1): 109–138.10.1007/BF02127700
  6. Brenner, S. and Scott, R. (2007). The Mathematical Theory of Finite Element Methods, Springer, New York, NY.10.1007/978-0-387-75934-0
  7. Cao, W., Zhang, X. and Zhang, Z. (2017). Superconvergence of immersed finite element methods for interface problems, Advances in Computational Mathematics43(4): 795–821.10.1007/s10444-016-9507-7
  8. Carstensen, C., Gallistl, D., Hellwing, F. and Weggler, L. (2014). Low-order DPG-FEM for an elliptic PDE, Computers & Mathematics with Applications68(11): 1503–1512.10.1016/j.camwa.2014.09.013
  9. Chen, Z. and Zou, J. (1998). Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik79(2): 175–202.10.1007/s002110050336
  10. Chou, S. (2012). An immersed linear finite element method with interface flux capturing recovery, Discrete and Continuous Dynamical Systems B17(7): 2343–2357.10.3934/dcdsb.2012.17.2343
  11. Chou, S.H., Kwak, D.Y. and Wee, K. (2010). Optimal convergence analysis of an immersed interface finite element method, Advances in Computational Mathematics33(2): 149–168.10.1007/s10444-009-9122-y
  12. Douglas Jr, J., Dupont, T. and Wheeler, M. (1974). A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems, Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique8(2): 47–59.
  13. Guo, H. and Yang, X. (2017). Gradient recovery for elliptic interface problem. II: Immersed finite element methods, Journal of Computational Physics338: 606–619.
  14. He, X., Lin, T. and Lin, Y. (2011). Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions, International Journal of Numerical Analysis and Modeling8(2): 284–301.
  15. Ji, H., Chen, J. and Li, Z. (2016). A new augmented immersed finite element method without using SVD interpolations, Numerical Algorithms71(2): 395–416.10.1007/s11075-015-9999-0
  16. Karczewska, A., Rozmej P., Szczeciński, M. and Boguniewicz, B. (2016). A finite element method for extended KdV equations, International Journal of Applied Mathematics and Computer Science26(3): 555–567, DOI: 10.1515/amcs-2016-0039.10.1515/amcs-2016-0039
  17. Kevorkian, J. (1990). Partial Differential Equations: Analytical Solution, Springer, New York, NY.10.1007/978-1-4684-9022-0
  18. Kwak, D.Y., Wee, K. and Chang, K. (2010). An analysis of a broken p1 nonconforming finite element method for interface problems, SIAM Journal on Numerical Analysis48(6): 2117–2134.10.1137/080728056
  19. Li, Z. (1998). The immersed interface method using a finite element formulation, Applied Numerical Mathematics27(3): 253–267.10.1016/S0168-9274(98)00015-4
  20. Li, Z. and Ito, K. (2006). The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, SIAM, Philadelphia, PA.10.1137/1.9780898717464
  21. Li, Z., Lin, T. and Wu, X. (2003). New Cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik96(1): 61–98.10.1007/s00211-003-0473-x
  22. Lin, T., Lin, Y. and Zhang, X. (2015). Partially penalized immersed finite element methods for elliptic interface problems, SIAM Journal on Numerical Analysis53(2): 1121–1144.10.1137/130912700
  23. Lin, T. and Zhang, X. (2012). Linear and bilinear immersed finite elements for planar elasticity interface problems, Journal of Computational and Applied Mathematics236(18): 4681–4699.10.1016/j.cam.2012.03.012
  24. Sutton, A. and Balluffi, R. (1995). Interfaces in Crystalline Materials, Clarendon Press, Oxford.
  25. Tartar, L. (2007). An Introduction to Sobolev Spaces and Interpolation Spaces, Springer, New York, NY.
  26. Wahlbin, L. (1995). Superconvergence in Galerkin Finite Element Methods, Springer, New York, NY.10.1007/BFb0096835
  27. Wheeler, M. (1974). A Galerkin procedure for estimating the flux for two-point boundary value problems, SIAM Journal on Numerical Analysis11(4): 764–768.10.1137/0711062
  28. Xu, J. (1982). Error estimates of the finite element method for the 2nd order elliptic equations with discontinuous coefficients, Journal of Xiangtan University1(1): 1–5.
  29. Yang, X., Li, B. and Li, Z. (2002). The immersed interface method for elasticity problems with interfaces, Dynamics of Continuous, Discrete and Impulsive Systems10(5): 783–808.
  30. Zhang, Z. and Naga, A. (2005). A new finite element gradient recovery method: Superconvergence property, SIAM Journal on Scientific Computing26(4): 1192–1213.10.1137/S1064827503402837
  31. Zienkiewicz, O. and Taylor, R. (2000). The Finite Element Method: Solid Mechanics, Butterworth-Heinemann, Oxford.
DOI: https://doi.org/10.1515/amcs-2017-0037 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Language: English
Page range: 527 - 537
Submitted on: Nov 9, 2016
Accepted on: May 31, 2017
Published on: Sep 23, 2017
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year

© 2017 Fangfang Qin, Zhaohui Wang, Zhijie Ma, Zhilin Li, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.