Skip to main content
×
×
Home

A Weak Formulation for Solving the Elliptic Interface Problems with Imperfect Contact

  • Liqun Wang (a1), Songming Hou (a2) and Liwei Shi (a3)
Abstract
Abstract

We propose a non-traditional finite element method with non-body-fitting grids to solve the matrix coefficient elliptic equations with imperfect contact in two dimensions, which has not been well-studied in the literature. Numerical experiments demonstrated the effectiveness of our method.

Copyright
Corresponding author
*Corresponding author. Email: wliqunhmily@gmail.com (L. Q. Wang), shou@latech.edu (S. M. Hou), sliweihmily@gmail.com (L. W. Shi)
References
Hide All
[1] Weisz J., On an iterative method for the solution of discretized elliptic problems with imperfect contact condition, J. Comput. Appl. Math., 72 (1996), pp. 319333.
[2] Barber J. R. and Zhang R., Transient behaviour and stability for the thermoelastic contact of the rods of dissimilar materials, Int. J. Mech. Sci., 30 (1988), pp. 691704.
[3] Wang L. and Shi L., Numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries, Int. J. PDE, Article ID 476873, (2013).
[4] Peskin C., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), pp. 220252.
[5] Peskin C. and Printz B., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys., 105 (1993), pp. 3346.
[6] Sussman M., Smereka P. and Osher S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), pp. 146154.
[7] Leveque R. J. and Li Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp. 10191044.
[8] Li Z. and Ito K., The Immersed Interface Method: Numerical Solutions of PDES Involving Interfaces and Irregular Domains, SIAM, Philadelphia, 2006.
[9] Li Z., A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35 (1998), pp. 230254.
[10] Liu X., Fedkiw R. P. and Kang M., A boundary condition capturing method for Poisson's equation on irregular domains, J. Comput. Phys., 160 (2000), pp. 151178.
[11] 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), pp. 457492.
[12] Wan J.W. L. and Liu X., A boundary condition capturing multigrid approach to irregular boundary problems, SIAM J. Sci. Comput., 25 (2004), pp. 19822003.
[13] Liu X. and Sideris T., Convergence of the ghost fluid method for elliptic equations with interfaces, Math. Comput., 72 (2003), pp. 17311746.
[14] Macklin P. and Lowengrub J. S., A new ghost cell/level set method for moving boundary problems: application to tumor growth, J. Sci. Comput., 35 (2008), pp. 266299.
[15] Li Z., Lin T., Lin Y. and Rogers R., An immersed finite element space and its approximation capability, Numer. Meth. PDE, 20 (2004), pp. 338367.
[16] Gong Y., Li B. and Li Z., Immersed-interface finite-element methods for elliptic interface problems with nonhomogeneous jump conditions, SIAM J. Numer. Anal., 46 (2008), pp. 472495.
[17] 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 (2011), pp. 284301.
[18] Lin T., Lin Y. and Zhang X., Partially penalized immersed finite element methods for elliptic interface problems, SIAM J. Numer. Anal., 53 (2015), pp. 11211144.
[19] Hou S. and Liu X., A numerical method for solving variable coefficient elliptic equations with interfaces, J. Comput. Phys., 202 (2005), pp. 411445.
[20] Hou S., Wang W. and Wang L., Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces, J. Comput. Phys., 229 (2010), pp. 71627179.
[21] Hou S., Li Z., Wang L. and Wang W., A numerical method for solving elasticity equations with interfaces, Commun. Comput. Phys., 12 (2012), pp. 595612.
[22] Hou S., Wang L. and Wang W., A numerical method for solving the elliptic interface problems with multi-domains and triple junction points, J. Comput. Math, 30 (2012), pp. 504516.
[23] Hou S., Song P., Wang L. and Zhao H., A weak formulation for solving elliptic interface problems without body fitted grid, J. Comput. Phys., 249 (2013), pp. 8095.
[24] Wang L., Hou S. and Shi L., A numerical method for solving 3D elasticity equations with sharp-edged interfaces, Int. J. PDE, Article ID 476873, (2013).
[25] Zhou Y., 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 (2006), pp. 130.
[26] Yu S., Zhou Y. and Wei G. W., Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces, J. Comput. Phys., 224 (2007), pp. 729756.
[27] Colella P. and Johansen H., A Cartesian grid embedded boundary method for Poisson's equation on irregular domains, J. Comput. Phys., 60 (1998), pp. 85147.
[28] Oevermann M. and Klein R., A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J. Comput. Phys., 219 (2006), pp. 749769.
[29] Oevermann M., Scharfenberg C. and Klein R., A sharp interface finite volume method for elliptic equations on Cartesian grids, J. Comput. Phys., 228 (2009), pp. 51845206.
[30] Chernogorova T., Ewing R. E., Iliev O. and Lazarov R., On the discretization of interface problems with perfect and imperfect contact, Lecture Notes in Physics, 552 (2000), pp. 93103, New York, Springer-Verlag.
[31] Li Z., Lin T. and Wu X., New cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik, 96 (2003), pp. 6198.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 69 *
Loading metrics...

Abstract views

Total abstract views: 264 *
Loading metrics...

* Views captured on Cambridge Core between 11th July 2017 - 18th January 2018. This data will be updated every 24 hours.