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Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver

Published online by Cambridge University Press:  03 June 2015

Wenqiang Feng ,
Xiaoming He ,
Yanping Lin  and
Xu Zhang
Wenqiang Feng*
Affiliation:
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
Xiaoming He*
Affiliation:
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
Yanping Lin*
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong Department of Mathematical and Statistics Science, University of Alberta, Edmonton, AB, T6G 2G1, Canada
Xu Zhang*
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
*
Email:[email protected]
Corresponding author.Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

This article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems. The numerical examples demonstrate the features of the proposed algorithms, including the optimal convergence in both L2 and semi-H1 norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location.

Keywords

65F1065N3035J1535K20Interface problemsimmersed finite elementsalgebraic multigrid method
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1045 - 1067
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Adjerid, S. and Lin, T., Higher-order immersed discontinuous Galerkin methods, Int. J. Inf. Syst. Sci., 3(4) (2007), 555–568.Google Scholar
[2]Adjerid, S. and Lin, T., p-th degree immersed finite element for boundary value problems with discontinuous coefficients, Appl. Numer. Math, 59(6) (2009), 1303–1321.Google Scholar
[3]Almgre, A., Bell, J. B., Collela, P., and Marthaler, T., Acartesian grid emthod for imcompressible Euler equations in complex geometries, SIAM J. Sci. Comput., 18 (1997), 1289–1390.Google Scholar
[4]Arakawa, Y. and Nakano, M., An efficient three-dimensional optics code for ion thruster research, In AIAA-3196, 1996.Google Scholar
[5]Babuška, I., The finite element method for elliptic equations with discontinuous coefficients, Computing, 5 (1970), 207–213.Google Scholar
[6]Babuška, I. and Osborn, J. E., Can a finite element method perform arbitrarily badly?, Math. Comput., 69(230) (2000), 443–462.Google Scholar
[7]Bendsøe, M. P. and Kikuchi, N., Generating optimal topologies in optimal design using a homogenization method, Comput. Meth. Appl. Mech. Eng., 71 (1988), 197–224.Google Scholar
[8]Boyd, I., VanGilde, D., and Liu, X., Monte carlo simulation of neutral xenon flows in electric propulsion devices, J. Propulsion Power, 14(6) (1998), 1009–1015.Google Scholar
[9]Bramble, J. H. and King, J. T., A finite element method for interface problems in domains with smooth boundary and interfaces, Adv. Comput. Math., 6 (1996), 109–138.Google Scholar
[10]Camp, B., Lin, T., Lin, Y., and Sun, W., Quadratic immersed finite element spaces and their approximation capabilities, Adv. Comput. Math., 24(1-4) (2006), 81–112.Google Scholar
[11]Cannon, J. R., Douglas, Jr, Jim, , and Denson Hill, C., A multi-boundary stefan problem and the disappearance of phases, J. Math. Mech., 17 (1967), 21–33.Google Scholar
[12]Chang, Q., Wong, Y., and Fu, H., On the algebraic multigrid method, J. Comput. Phys., 125 (1996), 279–292.Google Scholar
[13]Chen, T. and Strain, J., Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems, J. Comput. Phys., 227 (2008), 7503–7542.Google Scholar
[14]Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175–202.Google Scholar
[15]Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), 12.Google Scholar
[16]Chou, S., An immersed linear finite element method with interface flux capturing recovery, Discrete Contin. Dyn. Syst. Ser. B, 17(7) (2012), 2343–2357.Google Scholar
[17]Chou, S., Kwak, D. Y., and Wee, K. T., Optimal convergence analysis of an immersed interface finite element method, Adv. Comput. Math., 33(2) (2010), 149–168.Google Scholar
[18]Chu, Y., Cao, Y., He, X.-M., and Luo, M., Asymptotic boundary conditions for two-dimensional electrostatic field problems with immersed finite elements, Comput. Phys. Commun., 182(11) (2011), 2331–2338.Google Scholar
[19]Dadone, A. and Grossman, B., Design optimization of fluid dynamic problems using cartesian grids, In Srinivas, K., Armfield, S. and Morgan, P., editor, Computational Fluid Dynamics 2002, Verlin, 591–596, Springer.Google Scholar
[20]Ewing, R. E., Li, Z., Lin, T., and Lin, Y., The immersed finite volume element methods for the elliptic interface problems, Modelling ‘98 (prague), Math. Comput. Simul., 50(1-4) (1999), 63–76.CrossRefGoogle Scholar
[21]Feng, X. and Li, Z., Simplified immersed interface methods for elliptic interface problems with straight interfaces, Numer. Methods Partial Differential Equations, 28(1) (2012), 188–203.Google Scholar
[22]Feng, X., Li, Z., and Wang, L., Analysis and numerical methods for some crack problems, Int. J. Numer. Anal. Model. Ser. B, 2(2-3) (2011), 155–166.Google Scholar
[23]Fogelson, A. L. and Keener, J. P., Immersed interface methods for Neumann and related problems in two and three dimensions, SIAM J. Sci. Comput., 22 (2001), 1630–1654.Google Scholar
[24]Geng, W. and Wei, G. W., Multiscale molecular dynamics using the matched interface and boundary method, J. Comput. Phys., 230(2) (2011), 435–457.Google Scholar
[25]Gilding, B. H., Qualitative mathematical analysis of the Richards equation, Transport Porous Med., 6(5-6) (1991), 651–666.Google Scholar
[26]Gong, Y., Li, B., and Li, Z., Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions, SIAM J. Numer. Anal., 46 (2008), 472–495.Google Scholar
[27]Gong, Y. and Li, Z., Immersed interface finite element methods for elasticity interface problems with non-homogeneous jump conditions, Numer. Math. Theory Methods Appl., 3(1) (2010), 23–39.Google Scholar
[28]Griffith, B. E. and Lim, S., Simulating an elastic ring with bend and twist by an adaptive generalized immersed boundary method, Commun. Comput. Phys., 12(2) (2012), 433–461.Google Scholar
[29]Guy, R. D. and Philip, B., A multigrid method for a model of the implicit immersed boundary equations, Commun. Comput. Phys., 12(2) (2012), 378–400.Google Scholar
[30]He, X.-M., Bilinear Immersed Finite Elements for Interface Problems, Ph.D. dissertation, Virginia Polytechnic Institute and State University, 2009.Google Scholar
[31]He, X.-M., Lin, T., and Lin, Y., Approximation capability of a bilinear immersed finite element space, Numer. Methods Partial Differential Equations, 24(5) (2008), 1265–1300.Google Scholar
[32]He, X.-M., Lin, T., and Lin, Y., A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficients, Commun. Comput. Phys., 6(1) (2009), 185–202.Google Scholar
[33]He, X.-M., Lin, T., and Lin, Y., Interior penalty discontinuous Galerkin methods with bilinear IFE for a second order elliptic equation with discontinuous coefficient, dedicated to Professor David Russell’s 70th birthday, J. Syst. Sci. Complex., 23(3) (2010), 467–483.Google Scholar
[34]He, X.-M., Lin, T., and Lin, Y., Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions, Int. J. Numer. Anal. Model., 8(2) (2011), 284–301.Google Scholar
[35]He, X.-M., Lin, T., and Lin, Y., The convergence of the bilinear and linear immersed finite element solutions to interface problems, Numer. Methods Partial Differential Equations, 28(1) (2012), 312–330.Google Scholar
[36]He, X.-M., Lin, T., and Lin, Y., Aselective immersed discontinuous Galerkin method for elliptic interface problems, Math. Methods Appl. Sci., 2013, accepted.Google Scholar
[37]He, X.-M., Lin, T., Lin, Y., and Zhang, X., Immersed finite element methods for parabolic equations with moving interface, Numer. Methods Partial Differential Equations, 29(2) (2013), 619–646.Google Scholar
[38]Heinrich, B., Finite Difference Methods on Irregular Networks, volume 82 of Int. Series Numer. Math., Birkhäuser, Boston, 1987.Google Scholar
[39]Hewitt, D. W., The embedded curved boundary method for orthogonal simulation meshes, J. Comput. Phys., 138 (1997), 585–616.Google Scholar
[40]Ingram, D. M., Causon, D. M., and Mingham, C. G., Developments in Cartesian cut cell methods, Math. Comput. Simul., 61(3-6) (2003), s561–572.Google Scholar
[41]Ji, H., Lien, F.-S., and Yee, E., An efficient second-order accurate cut-cell method for solving the variable coeffcient Poisson equation with jump conditions on irregular domains, Int. J. Numer. Methods Fluids, 52 (2006), 723–748.Google Scholar
[42]Johansen, H. and Colella, P., A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. Comput. Phys., 147 (1998), 60–85.Google Scholar
[43]Kafafy, R., Lin, T., Lin, Y., and Wang, J., Three-dimensional immersed finite element methods for electric field simulation in composite materials, Int. J. Numer. Meth. Eng., 64(7) (2005), 940–972.Google Scholar
[44]Kafafy, R., Wang, J., and Lin, T., A hybrid-grid immersed-finite-element particle-in-cell simulation model of ion optics plasma dynamics, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 12 (2005), 1–16.Google Scholar
[45]Kim, Y., Seol, Y., Lai, M., and Peskin, C. S., The immersed boundary method for two-dimensional foam with topological changes, Commun. Comput. Phys., 12(2) (2012), 479–493.Google Scholar
[46]Kwak, D. Y., Wee, K. T., and Chang, K. S., An analysis of a broken p 1-nonconforming finite element method for interface problems, SIAM J. Numer. Anal., 48(6) (2010), 2117–2134.Google Scholar
[47]Chartier, T. P. and Adams, L., New geometric immersed interface multigrid solvers, SIAM J. Sci. Comput., 25(5) (2004), 1516–1533.Google Scholar
[48]Chartier, T. P. and Adams, L., A comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems, SIAM J. Sci. Comput., 26(3) (2005), 762–784.Google Scholar
[49]Li, Z. and Adams, L., The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput., 24(2) (2002), 463–479.Google Scholar
[50]LeVeque, R. J. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 34 (1994), 1019–1044.Google Scholar
[51]Li, Z., The immersed interface method using a finite element formulation, Appl. Numer. Math., 27(3) (1997), 253–267.Google Scholar
[52]Li, Z. and Ito, K., The immersed interface method: Numerical solutions of PDEs involving interfaces and irregular domains volume 33 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.Google Scholar
[53]Li, Z., Lin, T., Lin, Y., and Rogers, R. C., An immersed finite element space and its approximation capability, Numer. Methods Partial Differential Equations, 20(3) (2004), 338–367.Google Scholar
[54]Li, Z., Lin, T., and Wu, X., New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96(1) (2003), 61–98.Google Scholar
[55]Li, Z. and Song, P., An adaptive mesh refinement strategy for immersed boundary/interface methods, Commun. Comput. Phys., 12(2) (2012), 515–527.Google Scholar
[56]Lin, T., Lin, Y., Rogers, R. C., and Ryan, L. M., A rectangular immersed finite element method for interface problems, In Minev, P. and Lin, Y., editors, Advances in Computation: Theory and Practice, Vol. 7, 107–114. Nova Science Publishers, Inc., 2001.Google Scholar
[57]Lin, T., Lin, Y., and Sun, W., Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems, Discrete Contin. Dyn. Syst. Ser. B, 7(4) (2007), 807–823.Google Scholar
[58]Lin, T., Lin, Y., Sun, W., and Wang, Z., Immersed finite element methods for 4th order differential equations, J. Comput. Appl. Math., 235(13) (2011), 3953–3964.Google Scholar
[59]Lin, T. and Sheen, D., The immersed finite element method for parabolic problems with the Laplace transformation in time discretization, Int. J. Numer. Anal. Model., 10(2) (2013), 298–313.Google Scholar
[60]Lin, T. and Wang, J., An immersed finite element electric field solver for ion optics modeling, In Proceedings of AIAA Joint Propulsion Conference, Indianapolis, IN, July, 2002. AIAA, 2002–4263.Google Scholar
[61]Lin, T. and Wang, J., The immersed finite element method for plasma particle simulation, In Proceedings of AIAA Aerospace Sciences Meeting, Reno, NV, Jan., 2003, AIAA, 2003–0842.Google Scholar
[62]Meyer, G. H., Multidimensional stefan problems, SIAM J. Numer. Anal., 10 (1973), 522–538.Google Scholar
[63]Peskin, C. S., Flow patterns around heart valves, J. Comput. Phys., 10 (1972), 252–271.Google Scholar
[64]Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), 220–252.Google Scholar
[65]Ruge, J. W. and Stüben, K., Algebraic multigrid, In McCormick, S. F., editor, Multigrid methods, SIAM, Philadelphia, 4 (1987), 73–130.Google Scholar
[66]Sauter, S. A. and Warnke, R., Composite finite elements for elliptic boundary value problems with discontinuous coefficients, Computing, 77(1) (2006), 29–55.Google Scholar
[67]Stüben, K., Algebraic multigrid (AMG): experiences and comparisons, Appl. Math. Comput., 13 (1983), 419–451.Google Scholar
[68]Tan, Z., Le, D. V., Lim, K. M., and Khoo, B. C., Animmersed interface method for the simulation of inextensible interfaces in viscous fluids, Commun. Comput. Phys., 11(3) (2012), 925–950.Google Scholar
[69]Oosterlee, C.Trottenberg, U. and Schuller, A., Multigrid, volume 631, Academic Press, London, 2001.Google Scholar
[70]Vallaghe, S. and Papadopoulo, T., A trilinear immersed finite element method for solving the electroencephalography forward problem, SIAM J. Sci. Comput., 32(4) (2010), 2379–2394.Google Scholar
[71]Wang, J., He, X.-M., and Cao, Y., Modeling spacecraft charging and charged dust particle interactions on lunar surface, Proceedings of the 10th Spacecraft Charging Technology Conference, Biarritz, France, 2007.Google Scholar
[72]Wang, J., He, X.-M., and Cao, Y., Modeling electrostatic levitation of dusts on lunar surface, IEEE Trans. Plasma Sci., 36(5) (2008), 2459–2466.Google Scholar
[73]Wang, K., Wang, H., and Yu, X., An immersed eulerian-lagrangian localized adjoint method for transient advection-diffusion equations with interfaces, Int. J. Numer. Anal. Model., 9(1) (2012), 29–42.Google Scholar
[74]Wu, C., Li, Z., and Lai, M., Adaptive mesh refinement for elliptic interface problems using the non-conforming immersed finite element method, Int. J. Numer. Anal. Model., 8(3) (2011), 466–483.Google Scholar
[75]Xie, H., Li, Z., and Qiao, Z., A finite element method for elasticity interface problems with locally modified triangulations, Int. J. Numer. Anal. Model., 8(2) (2011), 189–200.Google Scholar
[76]Xu, S., An iterative two-fluid pressure solver based on the immersed interface method, Commun. Comput. Phys., 12(2) (2012), 528–543.Google Scholar
[77]Ye, M., Khaleel, R., and Yeh, T. J., Stochastic analysis of moisture plume dynamics of a field injection experiment, Water Resour. Res., 41 (2005), W03013.Google Scholar
[78]Zhao, S., High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces, J. Comput. Phys., 229(9) (2010), 3155–3170.Google Scholar
[79]Zhou, Y. C., Liu, J., and Harry, D. L., A matched interface and boundary method for solving multi-flow Navier-Stokes equations with applications to geodynamics, J. Comput. Phys., 231(1) (2012), 223–242.Google Scholar
[80]Zhou, Y. C. and Wei, G. W., On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method, J. Comput. Phys., 219(1) (2006), 228–246.Google Scholar
[81]Zhou, Y. C., Zhao, S., Feig, M., and 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