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A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs

Published online by Cambridge University Press:  03 June 2015

Wenjun Ying  and
Wei-Cheng Wang
Wenjun Ying*
Affiliation:
Department of Mathematics, MOE-LSC and Institute of Natural Sciences, Shanghai Jiao Tong University, Minhang, Shanghai 200240, P.R. China
Wei-Cheng Wang*
Affiliation:
Department of Mathematics, National Tsing Hua University, and National Center for Theoretical Sciences, HsinChu, 300, Taiwan
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.

Keywords

35J0565N0665N38Elliptic partial differential equationvariable coefficientskernel-free boundary integral methodfinite difference methodgeometric multigrid iteration
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1108 - 1140
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Adams, L., Chartier, T. P., New geometric immersed interface multigrid solvers, SIAM J. Sci. Comput. 25 (2004) 1516–1533.Google Scholar
[2]Aslam, T. D., A partial differential equation approach to multidimensional extrapolation, J. Comput. Phys. 193 (2003) 349–355.Google Scholar
[3]Beale, J. T., A grid-based boundary integral method for elliptic problems in three-dimensions, SIAM J. Numer. Anal. 42 (2004) 599–620.Google Scholar
[4]Beale, J. T., Lai, M. C., A method for computing nearly singular integrals, SIAM J. Numer. Anal. 38 (6) (2001) 1902–1925.Google Scholar
[5]Beale, J. T., Layton, A. T., On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. Appl. Math. Comput. Sci. 1 (2006) 91–119.Google Scholar
[6]Berthelsen, P. A., A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, J. Comput. Phys. 197 (2004) 364–386.Google Scholar
[7]Cahn, J. W., Hilliard, J. E., Free energy of a nonuniform system i, J. Chem. Phys. 28 (1958) 258–267.Google Scholar
[8]Calhoun, D., A Cartesian grid method for solving the two-dimensional stream function-vorticity equations in irregular regions, J. Comput. Phys. 176 (2002) 231–275.Google Scholar
[9]Chen, T., Strain, J., Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems, J. Comput. Phy. 227 (2008) 7503–7542.Google Scholar
[10]Cheng, H., Huang, J., Leiterman, T. J., An adaptive fast solver for the modified Helmholtz equation in two space dimensions, J. Comput. Phy. 211 (2) (2006) 616–637.Google Scholar
[11]Cheng, L. T., Fedkiw, R. P., Gibou, F., Kang, M., A second-order accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys. 171 (2001) 205–227.Google Scholar
[12]Chern, I. L., Shu, Y. C., A coupling interface method for elliptic interface problems, J. Comput. Phys. 225 (2007) 2138–2274.CrossRefGoogle Scholar
[13]Concus, P., Golub, G. H., Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations, SIAM J. Numer. Anal. 10 (1973) 1103–1120.Google Scholar
[14]Costabel, M., Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (3) (1988) 613–626.Google Scholar
[15]Elliott, C. M., Zheng, S., On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal. 96 (339).Google Scholar
[16]Englund, J., Helsing, J., A comparison of splittings and integral equation solvers for a non-separable elliptic equation, BIT Numerical Mathematics 44 (2004) 675–697.Google Scholar
[17]Fogelson, A. L., Keener, J. P., Immersed interface methods for Neumann and related problems in two and three dimensions, SIAM J. Sci. Comput. 22 (2000) 1630–1654.Google Scholar
[18]Gibou, F., Fedkiw, R., Cheng, L.-T., Kang, M., A second order accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys. 176 (2002) 205–227.Google Scholar
[19]Henriquez, C., Simulating the electrical behavior of cardiac tissue using the bidomain model, Critical Reviews in Biomedical Engineering 21 (1993) 1–77.Google ScholarPubMed
[20]Hou, S., Liu, X.-D., A numerical method for solving variable coefficient elliptic equation with interfaces, J. Comput. Phys. 202 (2005) 411–445.Google Scholar
[21]Huang, B., Tu, B., Lu, B., A fast direct solver for a class of 3-D elliptic partial differential equation with variable coefficient, Commun. Comput. Phys. 12 (2012) 1148–1162.Google Scholar
[22]Huang, J., Greengard, L., A fast direct solver for elliptic partial differential equations on adap-tively refined meshes, SIAM J. Sci. Comput. 21 (2000) 1551–1566.Google Scholar
[23]Johansen, H., Colella, P., A Cartesian grid embedding boundary method for Poisson’s equation on irregular domains, J. Comput. Phys. 147 (1998) 60–85.Google Scholar
[24]Lambers, J. V., A multigrid block Krylov subspace spectral method for variable coefficient elliptic PDE, AIP Conf. Proc. 1220 (2010) 134–149.Google Scholar
[25]LeVeque, R. J., Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. on Num. Anal. 31 (4) (1994) 1019–1044.Google Scholar
[26]Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal. 35 (1) (1998) 230–254.Google Scholar
[27]Li, Z., Ito, K., The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains, SIAM Frontiers in Applied Mathematics, 2006.Google Scholar
[28]Li, Z. L., Lubkin, S. R., Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension, Int. J. Numer. Meth. Fluid. 37 (2001) 525–540.CrossRefGoogle Scholar
[29]Li, Z. L., Wang, W.-C., Chern, I.-L., Lai, M.-C., New formulations for interface problems in polar coordinates, SIAM J. Sci. Comput. 25 (2003) 224–245.Google Scholar
[30]Linge, S., Sundnes, J., Hanslien, M., Lines, G. T., Tveito, A., Numerical solution of the bidomain equations, Phil. Trans. R. Soc. A 367 (2009) 1931–1950.Google Scholar
[31]Liu, X. D., Fedkiw, R. P., Kang, M., A boundary condition capturing method for Poisson’s equation on irregular domains, J. Comput. Phys. 160 (2000) 151–178.Google Scholar
[32]Lu, B., Zhou, Y., Holst, M., McCammon, J., Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications, Communications in Computational Physics 3 (5) (2008) 973–1009.Google Scholar
[33]Mayo, A., The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Numer. Anal. 21 (1984) 285–299.CrossRefGoogle Scholar
[34]Mayo, A., Fast high order accurate solution of Laplace’s equation on irregular regions, SIAM J. Sci. Statist. Comput. 6 (1985) 144–157.Google Scholar
[35]Min, C., Gibou, F., Ceniceros, H., A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids, J. Comput. Phys. 218 (2006) 123.Google Scholar
[36]Oevermann, M., Klein, R., A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J. Comput. Phys. 219 (2006) 749–769.Google Scholar
[37]Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys. 25 (1977) 220–252.CrossRefGoogle Scholar
[38]Peskin, C. S., Lectures on mathematical aspects of physiology, Lectures in Appl. Math. 19 (1981) 69–107.Google Scholar
[39]Peskin, C. S., The immersed boundary method, Acta Numer. (2002) 1–39.Google Scholar
[40]Proskurowski, W., Widlund, O., On the numerical solution of Helmholtz’s equation by the capacitance matrix method, Math. Comp. 30 (1976) 433–468.Google Scholar
[41]Fedkiw, B. M. R. P., Aslam, T., Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys. 152 (1999) 457–492.Google Scholar
[42]Saad, Y., Iterative methods for sparse linear systems, PWS Publishing Company, Boston, 1996.Google Scholar
[43]Saad, Y., Schultz, M. H., GMRES: A generalized minimal residual method for solving non-symmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856–869.Google Scholar
[44]Strain, J., Fast spectrally-accurate solution of variable-coefficient elliptic problems, Proc. American Math. Soc. 122 (3) (1994) 843–850.Google Scholar
[45]Vigmond, E. J., Santos, R. W. dos, Prassl, A. J., Deo, M., Plank, G., Solvers for the cardiac bidomain equations, Prog. Biophys Mol. Biol. 96 (2008) 3–18.Google Scholar
[46]Wiegmann, A., Bube, K. P., The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal. 37 (3) (2000) 827–862.Google Scholar
[47]Ying, W.-J., Beale, J. T., A fast accurate boundary integral method for potentials on closely packed cells, Communications in Computational Physics 14 (4) (2013) 1073–1093.Google Scholar
[48]Ying, W.-J., Henriquez, C. S., A kernel-free boundary integral method for elliptic boundary value problems, J. Comp. Phy. 227 (2) (2007) 1046–1074.Google Scholar
[49]Ying, W.-J., Wang, W.-C., A kernel-free boundary integral method for implicitly defined surfaces, Journal of Computational Physics 252 (2013) 606–624.Google Scholar
[50]Zhou, S., Wang, Z., Li, B., Mean-field description of ionic size effects with non-uniform ionic sizes: a numerical approach, Phys. Rev. E 84 (2011) 021901.Google Scholar
[51]Zhou, Y. C., Zhao, S., Feig, M., Wei, G. W., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys. 213 (1) (2006) 1–30.Google Scholar
[52]Zhu, J., Chen, L. Q., Shen, J., Tikare, V., Onuki, A., Coarsening kinetics from a variable mobility Cahn-Hilliard equation: Application of a semi-implicit Fourier spectral method, Phys. Rev. E 60 (1999) 3564.Google Scholar