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 & 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. doi: 10.1137/S0036142999361517.

[2]

X. He, "Bilinear Immersed Finite Elements for Interface Problems,", Ph.D thesis, (2009).

[3]

Z. Li, The immersed interface method using a finite element formulation,, Applied Numerical Mathemtics, 27 (1998), 253. 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 (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. 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. 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, 7 (2001), 107.

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. doi: 10.1137/S0036142999361517.

[2]

X. He, "Bilinear Immersed Finite Elements for Interface Problems,", Ph.D thesis, (2009).

[3]

Z. Li, The immersed interface method using a finite element formulation,, Applied Numerical Mathemtics, 27 (1998), 253. 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 (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. 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. 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, 7 (2001), 107.

[1]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[14]

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

[15]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927

[16]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[17]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[18]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[19]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[20]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]