Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions

SIAM Journal on Numerical Analysis

Volumn 46, Issue 1, 2007, Pages 472-495

  Gong, Yan ^a   Li, Bo ^b   Li, Zhilin ^a  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Elliptic interface problems; Error estimates; Immersed interface finite element method; Level set functions; Nonhomogeneous jump conditions

Indexed keywords

ELLIPTIC INTERFACE PROBLEMS; ERROR ESTIMATES; IMMERSED-INTERFACE FINITE-ELEMENT METHOD; LEVEL-SET FUNCTIONS; NONHOMOGENEOUS JUMP CONDITIONS;

CONVERGENCE OF NUMERICAL METHODS; ERROR ANALYSIS; FINITE ELEMENT METHOD; NUMERICAL METHODS; PROBABILITY DENSITY FUNCTION; SET THEORY;

PROBLEM SOLVING;

EID: 55549138751     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060666482     Document Type: Article

References (23)

1. L. ADAMS AND Z. LI, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput., 24 (2002), pp. 463-479.
2. R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
3. E. BÄNSCH, F. HAUßER, O. LAKKIS, B. LI, AND A. VOIGT, Finite element method for epitaxial growth with attachment-detachment kinetics, J. Comput. Phys., 194 (2004), pp. 409-434.
4. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, 2nd ed., Springer-Verlag, New York, 2002.
5. W. K. BURTON, N. CABRERA, AND F. C. FRANK, The growth of crystals and the equilibrium of their surfaces, Phil. Trans. Roy. Soc. London Ser. A, 243 (1951), pp. 299-358.
6. R. E. CAFLISCH AND B. LI, Analysis of island dynamics in epitaxial growth of thin films, Multiscale Model. Simul., 1 (2003), pp. 150-171.
7. Z. CHEN AND J. ZOU, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), pp. 175-202.
8. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
9. A. DEMLOW AND G. DZIUK, An adaptive finite element method for the Laplace-Beltrami operator on implicity defined surfaces, SIAM J. Numer. Anal., 45 (2007), pp. 421-442.
10. P. GREMAUD, C. KUSTER, AND Z. LI, A study of numerical methods for the level set approach, Appl. Numer. Math., 57 (2007), pp. 837-846.
11. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program, Boston, MA, 1985.
12. S.-M. HOU AND X.-D. LIU, A numerical method for solving variable coefficient elliptic equation with interfaces, J. Comput. Phys., 202 (2005), pp. 411-445.
13. J. N. ISRAELACHVILI, Intermolecular and Surface Forces with Applications to Colloidal and Biological Systems, 2nd ed., Academic Press, New York, 1990.
14. C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, UK, 1987.
15. Z. LI, A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35 (1998), pp. 230-254.
16. Z. LI AND K. ITO, Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23 (2001), pp. 339-361.
17. Z. LI AND K. ITO, The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, Frontiers Appl. Math. 33, SIAM, Philadelphia, 2006.
18. Z. LI, T. LIN, AND X. WU, New Cartesian grid methods for interface problem using finite element formulation, Numer. Math., 96 (2003), pp. 61-98.
19. Z. LI, W.-C. WANG, I.-L. CHERN, AND M.-C. LAI, New formulations for interface problems in polar coordinates, SIAM J. Sci. Comput., 25 (2003), pp. 224-245.
20. S. OSHER AND R. FEDKIW, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag,
21. S. OSHER AND J. SETHIAN, Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), pp. 12-49.
22. J. A. SETHIAN, Level Set Methods and Fast Marching Methods, 2nd ed., Cambridge University Press, Cambridge, UK, 1999.
23. D. YOUNG AND D. KINCAID, The ITPACK package for large sparse linear systems, in Elliptic Problem Solvers, M. Schultz et al., eds., Academic Press, New York, 1981, pp. 163-185.