October  2012, 17(7): 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

An immersed linear finite element method with interface flux capturing recovery

1. 

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43402-0221, United States

Received  May 2011 Revised  April 2012 Published  July 2012

A flux recovery technique is introduced for the computed solution of an immersed finite element method for one dimensional second-order elliptic problems. The recovery is by a cheap formula evaluation and is carried out over a single element at a time while ensuring the continuity of the flux across the interelement boundaries and the validity of the discrete conservation law at the element level. Optimal order rates are proved for both the primary variable and its flux. For piecewise constant coefficient problems our method can capture the flux at nodes and at the interface points exactly. Moreover, it has the property that errors in the flux are all the same at all nodes and interface points for general problems. We also show second order pressure error and first order flux error at the nodes. Numerical examples are provided to confirm the theory.
Citation: So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343
References:
[1]

S.-H. Chou and S. Tang, Conservative $P1$ conforming and nonconforming Galerkin FEMs: Effective flux evaluation via a nonmixed method approach, SIAM J. Numer. Anal., 38 (2000), 660-680. doi: 10.1137/S0036142999361517.

[2]

X. He, "Bilinear Immersed Finite Elements for Interface Problems," Ph.D thesis, Virginia Tech., Blacksberg, VA, 2009.

[3]

Z. Li, The immersed interface method using a finite element formulation, Applied Numerical Mathemtics, 27 (1998), 253-267. doi: 10.1016/S0168-9274(98)00015-4.

[4]

Z. Li and K. Ito, "The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains," Frontiers in Applied Mathematics, 33, SIAM, Philadelphia, PA, 2006.

[5]

Z. Li, T. Lin, Y. Lin and R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Methods. Partial Differential Equation, 20 (2004), 338-367. doi: 10.1002/num.10092.

[6]

Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96 (2003), 61-98. doi: 10.1007/s00211-003-0473-x.

[7]

T. Lin, Y. Lin, R. Rogers and M. L. Ryan, A rectangular immersed finite element space for interface problems, in "Scientific Computing and Applications" (Kananaskis, AB, 2000), Advances in Computation: Theory and Practice, 7, Nova Sci. Publ., Huntington, NY, (2001), 107-114.

show all references

References:
[1]

S.-H. Chou and S. Tang, Conservative $P1$ conforming and nonconforming Galerkin FEMs: Effective flux evaluation via a nonmixed method approach, SIAM J. Numer. Anal., 38 (2000), 660-680. doi: 10.1137/S0036142999361517.

[2]

X. He, "Bilinear Immersed Finite Elements for Interface Problems," Ph.D thesis, Virginia Tech., Blacksberg, VA, 2009.

[3]

Z. Li, The immersed interface method using a finite element formulation, Applied Numerical Mathemtics, 27 (1998), 253-267. doi: 10.1016/S0168-9274(98)00015-4.

[4]

Z. Li and K. Ito, "The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains," Frontiers in Applied Mathematics, 33, SIAM, Philadelphia, PA, 2006.

[5]

Z. Li, T. Lin, Y. Lin and R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Methods. Partial Differential Equation, 20 (2004), 338-367. doi: 10.1002/num.10092.

[6]

Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96 (2003), 61-98. doi: 10.1007/s00211-003-0473-x.

[7]

T. Lin, Y. Lin, R. Rogers and M. L. Ryan, A rectangular immersed finite element space for interface problems, in "Scientific Computing and Applications" (Kananaskis, AB, 2000), Advances in Computation: Theory and Practice, 7, Nova Sci. Publ., Huntington, NY, (2001), 107-114.

[1]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807

[2]

Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323

[3]

Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175

[4]

Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373

[5]

Ben A. Vanderlei, Matthew M. Hopkins, Lisa J. Fauci. Error estimation for immersed interface solutions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1185-1203. doi: 10.3934/dcdsb.2012.17.1185

[6]

Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89

[7]

Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, 2021, 29 (5) : 3171-3191. doi: 10.3934/era.2021032

[8]

Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838

[9]

Na Peng, Jiayu Han, Jing An. An efficient finite element method and error analysis for fourth order problems in a spherical domain. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022021

[10]

Harvey A. R. Williams, Lisa J. Fauci, Donald P. Gaver III. Evaluation of interfacial fluid dynamical stresses using the immersed boundary method. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 519-540. doi: 10.3934/dcdsb.2009.11.519

[11]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[12]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387

[13]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[14]

Xiaoxiao He, Fei Song, Weibing Deng. A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2849-2871. doi: 10.3934/dcdsb.2021163

[15]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089

[16]

Zhongyi Huang. Tailored finite point method for the interface problem. Networks and Heterogeneous Media, 2009, 4 (1) : 91-106. doi: 10.3934/nhm.2009.4.91

[17]

Robert H. Dillon, Jingxuan Zhuo. Using the immersed boundary method to model complex fluids-structure interaction in sperm motility. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 343-355. doi: 10.3934/dcdsb.2011.15.343

[18]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349

[19]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[20]

Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]