# American Institute of Mathematical Sciences

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.

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##### 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.
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