Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T11:02:09.164Z Has data issue: false hasContentIssue false
  • English
  • Français

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

Published online by Cambridge University Press:  03 June 2015

Tao Lin,
Yanping Lin and
Xu Zhang
Tao Lin*
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
Yanping Lin*
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong KongDepartment 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: tlin@math.vt.edu
Corresponding author. Email: malin@polyu.edu.hk
Email: xuz@vt.edu
Get access

Abstract

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

Keywords

35R0565M2065M60Immersed finite elementmoving interfacemethod of linesCartesian mesh

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013, pp. 548 - 568
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Ascher, U. and Petzold, L., Computer methods for ordinary differential equations and differential-algebraic equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.Google Scholar
[2]Babuška, I., The finite element method for elliptic equations with discontinuous coefficients, Computing, 5 (1970), pp. 207–213.Google Scholar
[3]Babuška, I., Caloz, C., and Osborn, J., Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 31(4) (1994), pp. 945–981.Google Scholar
[4]Babuška, I. and Osborn, J., Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal., 20(3) (1983), pp. 510–536.Google Scholar
[5]Babuška, I. and Osborn, J., Can a finite element method perform arbitrarily badly?, Math. Comput., 69(230) (2000), pp. 443–462.Google Scholar
[6]Barrett, J. and Elliott, C., Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA J. Numer. Anal., 7(3) (1987), pp. 283–300.CrossRefGoogle Scholar
[7]Bramble, J. and King, J., A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math., 6(2) (1996), pp. 109–138.Google Scholar
[8]Butcher, J., Numerical methods for ordinary differential equations, John Wiley & Sons, Ltd., Chichester, second edition, 2008.Google Scholar
[9]Cannon, J., Douglas, J., and Hill, C., A multi-boundary Stefan problem and the disappearance of phases, J. Math. Mech., 17 (1967), pp. 21–33.Google Scholar
[10]Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79(2) (1998), pp. 175–202.CrossRefGoogle Scholar
[11]Chou, S., Kwak, D., and Wee, K., Optimal convergence analysis of an immersed interface finite element method, Adv. Comput. Math., 33(2) (2010), pp. 149–168.Google Scholar
[12]Chu, C., Graham, I., and Hou, T., A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comput., 79(272) (2010), pp. 1915–1955.CrossRefGoogle Scholar
[13]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(2) (1999), pp. 457–492.Google Scholar
[14]Feng, X., Li, Z. and Wang, L., Analysis and numerical methods for some crack problems, Int. J. Numer. Anal. Model. Series B, 2(2-3) (2011), pp. 155–166.Google Scholar
[15]Gilding, B., Qualitative mathematical analysis of the Richards equation, Transport Porous Med., 6(5-6) (1991), pp. 651–666.Google Scholar
[16]Hairer, E. and Wanner, G., Solving ordinary differential equations, volume II of Springer Series in Computational Mathematics, 14, Springer-Verlag, Berlin, 1996.Google Scholar
[17]Hansbo, A. and Hansbo, P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., 191 (2002), pp. 5537–5552.Google Scholar
[18]Hansbo, A. and Hansbo, P., A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Comput. Methods Appl. Mech. Eng., 193 (2004), pp. 3523–3540.Google Scholar
[19]He, X., Bilinear Immersed Finite Elements For Interface Problems, PhD thesis, Virginia Tech, 2009.Google Scholar
[20]He, X., Lin, T., and Lin, Y., Approximation capability of a bilinear immersed finite element space, Numer. Methods Partial Differential Equations, 24(5) (2008), pp. 1265–1300.Google Scholar
[21]He, X., 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), pp. 284–301.Google Scholar
[22]He, X., 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
[23]Leveque, R. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31(4) (1994), pp. 1019–1044.Google Scholar
[24]Li, Z., Immersed interface methods for moving interface problems, Numer. Algorithms, 14(4) (1997), pp. 269–293.Google Scholar
[25]Li, Z., The immersed interface method using a finite element formulation, Appl. Numer. Math., 27(3) (1998), pp. 253–267.CrossRefGoogle Scholar
[26]Li, Z. and Ito, K., The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains, Frontiers in Applied Mathematics, 33, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.Google Scholar
[27]Li, Z., Lin, T., Lin, Y., and Rogers, R., An immersed finite element space and its approximation capability, Numer. Methods Partial Differential Equations, 20(3) (2004), pp. 338–367.Google Scholar
[28]Li, Z., Lin, T., and Wu, X., New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96(1) (2003), pp. 61–98.Google Scholar
[29]Lin, T., Lin, Y., Rogers, R., and Ryan, M., A rectangular immersed finite element space for interface problems, Adv. Comput. Theory Pract., 7 (2001), pp. 107–114.Google Scholar
[30]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), pp. 298–313.Google Scholar
[31]Lin, T. and Zhang, X., Linear and bilinear immersed finite elements for planar elasticity interface problems, J. Comput. Appl. Math., 236(18) (2012), pp. 4681–4699.Google Scholar
[32]Liu, X., Fedkiw, R., and Kang, M., A boundary condition capturing method for Poisson’s equation on irregular domains, J. Comput. Phys., 160(1) (2000), pp. 151–178.Google Scholar
[33]Marheineke, N., Liljo, J., Mohring, J., Schnebele, J. and Wegener, R., Multiphysics and multimethods problem of rotational glass fiber melt-spinning, Int. J. Numer. Anal. Model. Series B, 3(3) (2012), pp. 330–344.Google Scholar
[34]Meyer, H., Multidimensional Stefan problems, SIAM J. Numer. Anal., 10 (1973), pp. 522–538.Google Scholar
[35]Noutcheuwa, R. and Owen, R. G., A new incompressible smoothed particle hydrodynamics-immersed boundary method, Int. J. Numer. Anal. Model. Series B, 3(2) (2012), pp. 126–167.Google Scholar
[36]Reddy, S. and Trefethen, L., Stability of the method of lines, Numer. Math., 62(2) (1992), pp. 235–267.Google Scholar
[37]Schiesser, W. E. and Griffiths, G. W., A compendium of partial differential equation models, Method of Lines Analysis with Matlab, Cambridge University Press, Cmbridge, 2009.Google Scholar
[38]Thomee, V., Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, second edition, 2006.Google Scholar
[39]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), pp. 29–42.Google Scholar
[40]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), pp. 466–483.Google Scholar
[41]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), pp. 189–200.Google ScholarPubMed
[42]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.CrossRefGoogle Scholar
[43]Zafarullah, A., Application of the method of lines to parabolic partial differential equations with error estimates, J. ACM, 17(2) (1970), pp. 294–302.Google Scholar
[44]Zhou, Y. and Wei, G., On the fictitious-domain and interpolation formulations of the matched interface and boundary (mib) method, J. Comput. Phys., 219(1) (2006), pp. 228–246.CrossRefGoogle Scholar
[45]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(1) (2006), pp. 1–30.CrossRefGoogle Scholar