An interpolation matched interface and boundary method for elliptic interface problems

Journal of Computational and Applied Mathematics

Volumn 234, Issue 1, 2010, Pages 73-94

  Pan, Kejia ^a,b   Tan, Yongji ^b   Hu, Hongling ^c  

^a CENTRAL SOUTH UNIVERSITY   (China)

^b FUDAN UNIVERSITY   (China)

^c HUNAN NORMAL UNIVERSITY   (China)

Author keywords

Discrete maximum principle; Elliptic interface problems; Finite difference scheme; Ghost fluid method; Immersed interface method; Matched interface and boundary

Indexed keywords

DISCRETE MAXIMUM PRINCIPLES; ELLIPTIC INTERFACE PROBLEMS; FINITE DIFFERENCE SCHEME; GHOST FLUID METHOD; IMMERSED INTERFACE METHOD;

COMPRESSIBLE FLOW; ERROR ANALYSIS; FUNCTIONS; GEOMETRY; INTERPOLATION; MAXIMUM PRINCIPLE; SET THEORY;

FINITE DIFFERENCE METHOD;

EID: 77649188716     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2009.12.006     Document Type: Article

References (38)

1. Babuska I. The finite element method for elliptic equations with discontinuous coeffcients. Computing 5 (1970) 207-213
2. Chen Z., and Zou J. Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79 (1998) 175-202
3. Griffith B., and Peskin C. On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems. J. Comput. Phys. 208 (2005) 75-105
4. Lai M., and Peskin C. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys. 160 (2000) 705-719
5. Peskin C. Numerical analysis of blood flow in heart. J. Comput. Phys. 25 (1977) 220-252
6. Peskin C. Lectures on mathematical aspects of physiology. Lect. Appl. Math. 19 (1981) 69-109
7. Peskin C., and McQueen D. A 3-dimensional computational method for blood-flow in the heart. 1. immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. 81 (1989) 372-405
8. Peskin C., and Printz B. Improved volume conservation in the computation of flows with immersed elastic boundaries. J. Comput. Phys. 105 (1993) 33-46
9. Adams L., and Li Z. The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. 24 (2002) 463-479
10. Deng S., Ito K., and Li Z. Three-dimensional elliptic solvers for interface problems and applications. J. Comput. Phys. 184 (2003) 215-243
11. Sethian J., and Wiegmann A. Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163 (2000) 489-528
12. Leveque R., and Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (1994) 1019-1044
13. Li Z., and Ito K. Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput. 23 (2001) 339-361
14. Li Z. An overview of the immersed interface method and its application. Taiwan. J. Math. 7 (2003) 1-49
15. Li Z. A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35 (1998) 230-254
16. Wiegmann A., and Bube K. The explicit-jump immersed interface method: Finite difference methods for pdes with piecewise smooth solutions. SIAM J. Numer. Anal. 37 (2000) 827-862
17. Li Z. The immersed interface method using a finite element formulation. Appl. Numer. Math. 27 (1998) 253-267
18. Hou S., and Liu X. A numerical method for solving variable cofficient elliptic equation with interfaces. J. Comput. Phys. 202 (2005) 411-445
19. Huang H., and Li Z. Convergence analysis of the immersed interface method. IMA J. Numer. Anal. 19 (1999) 583-608
20. Fedkiw R., Aslam T., Merriman B., and Osher S. A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (1999) 457-492
21. Liu X., Fedkiw R., and Kang M. A boundary condition capturing method for poissons equation on irregular domains. J. Comput. Phys. 160 (2000) 151-178
22. Kang M., Fedkiw R., and Liu X. A boundary condition capuring method for multiphase incompressible flow. J. Sci. Comput. 15 (2000) 323-360
23. Yu S., Zhou Y., and Wei G. Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys. 224 (2007) 729-756
24. Yu S., and Wei G. Three-dimensional mathed interface and boundary (MIB) method for treating geometric singularities. J. Comput. Phys. 227 (2007) 602-623
25. Zhao S., and Wei G. High order FDTD methods via derivative matching for maxwell equations with material interfaces. J. Comput. Phys. 200 (2004) 60-103
26. Zhou Y., Zhao S., Feig M., and Wei G. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213 (2006) 1-30
27. Zhou Y., and Wei G. On the fictious-domain and interpolation formulations of the matched interface and boundary method. J. Comput. Phys. 219 (2006) 228-246
28. Bramble J., and King J. A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6 (1996) 109-138
29. Li Z., Lin T., et al. New cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96 (2003) 61-98
30. Liu W., Liu Y., et al. Immersed finite element method and its applications to biological systems. Comput. Methods Appl. Mech. Eng. 195 (2006) 1722-1749
31. Sinha R., and Deka B. On the convergence of finite element method for second order elliptic interface problems. Numer. Funct. Anal. Optim. 27 (2006) 99-115
32. Huang J., and Zou J. A mortar element method for elliptic problems with discontinuous coefficients. IMA J. Numer. Anal. 22 (2002) 549-576
33. Wang W. A jump condition capturing finite difference scheme for elliptic interface problems. SIAM J. Sci. Comput. 25 (2004) 1479-1496
34. Li Z., Wang W., Chern I., and Lai M. New formulations for interface problems in polar coordinates. SIAM J. Sci. Comput. 25 (2003) 224-245
35. Yu S., Geng W., and Wei G. Treatment of geometric singularities in the implicit solvent models. J. Chem. Phys. 126 (2007) 244108
36. Liu X., and Sideris T.C. Convergence of the ghost fluid method for elliptic equations with interfaces. Math. Comput. 72 (2003) 1731-1746
37. Fornberg B. Calculation of weights in finite difference formulas. SIAM Rev. 40 (1998) 685-691
38. Ciarlet P. Discrete maximum principles for finite difference operators. Aequationes Math. 4 (1970) 336-352