Abstract
The augmented strategies for interface problems and problems defined on irregular domains are reviewed in this paper. There are at least two reasons to use augmented strategies. The first one is to get faster algorithms, particularly, to take advantages of existing fast solvers. The second reason is that, for some interface problems, an augmented approach may be the only way to derive an accurate algorithm. Using an augmented approach, one or several quantities of co-dimension one are introduced. The GMRES iterative method is often used to solve the augmented variable(s) that are only defined along the interface or the irregular boundary. Several examples of augmented methods are provided in this paper.
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References
Adams, J., Swarztrauber, P., Sweet, R.: Fishpack, http://www.netlib.org/fishpack/
Chen, G.: Immersed interface method for biharmonic equations defined on irregular domains and its application to the Stokes flow, Ph.D thesis. North Carolina State University (2003)
Chen, G., Li, Z., Lin, P.: A fast finite difference method for biharmonic equations on irregular domains. CRSC-TR04-09, North Carolina State University (2004)
Cortez, R.: The method of regularized Stokeslets. SIAM J. Sci. Comput. 23, 1204–1225 (2001)
Deng, S., Ito, K., Li, Z.: Three dimensional elliptic solvers for interface problems and applications. J. Comput. Phys. 184, 215–243 (2003)
Dumett, M., Keener, J.: A numerical method for solving anisotropic elliptic boundary value problems on an irregular domain in 2D. SIAM J. Sci. Comput. (2002) (in press)
Fogelson, A.L., Keener, J.P.: Immersed interface methods for Neumann and related problems in two and three dimensions. SIAM J. Sci. Comput. 22, 1630–1684 (2000)
Fogelson, A.L., Peskin, C.S.: Numerical solution of the three dimensional Stokes equations in the presence of suspended particles. In: Proc. SIAM Conf. Multi-phase Flow. SIAM, Philadelphia (1986)
Greenbaum, A., Greengard, L., Mayo, A.: On the numerical-solution of the biharmonic equation in the plane. PHYSICA D 60, 216–225 (1992)
Hou, T., Li, Z., Osher, S., Zhao, H.: A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys. 134, 236–252 (1997)
Hunter, J., Li, Z., Zhao, H.: Autophobic spreading of drops. J. Comput. Phys. 183, 335–366 (2002)
Ito, K., Li, Z.: Interface relations for Stokes equations with discontinuous viscosity and singular sources (preprint) (2004)
Johansen, H., Colella, P.: A Cartesian grid embedded boundary method for Poisson equations on irregular domains. J. Comput. Phys. 147, 60–85 (1998)
LeVeque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)
LeVeque, R.J., Li, Z.: Immersed interface method for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18, 709–735 (1997)
Li, Z.: The Immersed Interface Method — A Numerical Approach for Partial Differential Equations with Interfaces. PhD thesis, University of Washington (1994)
Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35, 230–254 (1998)
Li, Z.: An overview of the immersed interface method and its applications. Taiwanese J. Mathematics 7, 1–49 (2003)
Li, Z.: IIMPACK, a collection of fortran codes for interface problems. Anonymous ftp at ftp.ncsu.edu under the directory: /pub/math/zhilin/Package, last updated (2001)
Li, Z., Ito, K.: Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput. 23, 1225–1242 (2001)
Li, Z., Ito, K., Lai, M.-C.: An augmented approach for stokes equations with discontinuous viscosity and singular forces. NCSU-CRSC Tech. Report: CRSC-TR04-23, North Carolina State Univeristy (2004)
Li, Z., Zhao, H., Gao, H.: A numerical study of electro-migration voiding by evolving level set functions on a fixed cartesian grid. J. Comput. Phys. 152, 281–304 (1999)
Mayo, A.: The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21, 285–299 (1984)
Mayo, A., Greenbaum, A.: Fast parallel iterative solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Sci. Stat. Comput. 13, 101–118 (1992)
Mayo, A., Peskin, C.S.: An implicit numerical method for fluid dynamics problems with immersed elastic boundaries. Contemp. Math. 141, 261–277 (1991)
McKenney, A., Greengard, L., Mayo, A.: A fast poisson solver for complex geometries. J. Comput. Phys. 118 (1995)
Proskurowski, W., Widlund, O.: On the numerical solution of Helmholtz’s equation by the capacitance matrix method. Math. Comp. 30, 433–468 (1976)
Saad, Y.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)
Tu, C., Peskin, C.S.: Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods. SIAM J. Sci. Stat. Comput. 13, 1361–1376 (1992)
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Li, Z. (2005). Augmented Strategies for Interface and Irregular Domain Problems. In: Li, Z., Vulkov, L., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2004. Lecture Notes in Computer Science, vol 3401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31852-1_7
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