Close Search Advanced
Journals
Resources
About Us
Open Access

Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver

Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver
Preview
Add to basket

Year:    2014

Communications in Computational Physics, Vol. 15 (2014), Iss. 4 : pp. 1045–1067

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 is 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 $L^2$ and semi-$H^1$ 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.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.150313.171013s

Communications in Computational Physics, Vol. 15 (2014), Iss. 4 : pp. 1045–1067

Published online:    2014-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:   

  1. A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma–lunar surface interactions

    Han, Daoru | Wang, Pu | He, Xiaoming | Lin, Tao | Wang, Joseph

    Journal of Computational Physics, Vol. 321 (2016), Iss. P.965

    https://doi.org/10.1016/j.jcp.2016.05.057 [Citations: 51]

  2. An iterative immersed finite element method for an electric potential interface problem based on given surface electric quantity

    Cao, Yong | Chu, Yuchuan | He, Xiaoming | Lin, Tao

    Journal of Computational Physics, Vol. 281 (2015), Iss. P.82

    https://doi.org/10.1016/j.jcp.2014.10.014 [Citations: 23]

  3. Phase field smoothing-PINN: A neural network solver for partial differential equations with discontinuous coefficients

    He, Rui | Chen, Yanfu | Yang, Zihao | Huang, Jizu | Guan, Xiaofei

    Computers & Mathematics with Applications, Vol. 171 (2024), Iss. P.188

    https://doi.org/10.1016/j.camwa.2024.07.024 [Citations: 0]

  4. Modeling and an immersed finite element method for an interface wave equation

    Bai, Jinwei | Cao, Yong | He, Xiaoming | Liu, Hongyan | Yang, Xiaofeng

    Computers & Mathematics with Applications, Vol. 76 (2018), Iss. 7 P.1625

    https://doi.org/10.1016/j.camwa.2018.07.015 [Citations: 22]

  5. Multigrid Algorithm for Immersed Finite Element Discretizations of Elliptic Interface Problems

    Chu, Hanyu | Song, Yongzhong | Ji, Haifeng | Cai, Ying

    Journal of Scientific Computing, Vol. 98 (2024), Iss. 1

    https://doi.org/10.1007/s10915-023-02416-x [Citations: 0]

  6. An adaptive immersed finite element method for linear parabolic interface problems with nonzero flux jump

    Ray, Tanushree | Sinha, Rajen Kumar

    Calcolo, Vol. 60 (2023), Iss. 2

    https://doi.org/10.1007/s10092-023-00515-7 [Citations: 0]

  7. Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

    Wang, Yang | Chen, Yanping | Huang, Yunqing | Liu, Ying

    Applied Mathematics and Mechanics, Vol. 40 (2019), Iss. 11 P.1657

    https://doi.org/10.1007/s10483-019-2538-7 [Citations: 15]

  8. Meshless analysis of elliptic interface boundary value problems

    Ahmad, Masood

    Engineering Analysis with Boundary Elements, Vol. 92 (2018), Iss. P.38

    https://doi.org/10.1016/j.enganabound.2017.07.008 [Citations: 19]

  9. Immersed finite element methods for unbounded interface problems with periodic structures

    Cao, Yong | Chu, Yuchuan | Zhang, Xiaoshi | Zhang, Xu

    Journal of Computational and Applied Mathematics, Vol. 307 (2016), Iss. P.72

    https://doi.org/10.1016/j.cam.2016.04.020 [Citations: 31]

  10. An implicit particle-in-cell model based on anisotropic immersed-finite-element method

    Bai, Jinwei | Cao, Yong | He, Xiaoming | E, Peng

    Computer Physics Communications, Vol. 261 (2021), Iss. P.107655

    https://doi.org/10.1016/j.cpc.2020.107655 [Citations: 13]

  11. A robust multigrid method for one dimensional immersed finite element method

    Wang, Saihua | Wang, Feng | Xu, Xuejun

    Numerical Methods for Partial Differential Equations, Vol. 37 (2021), Iss. 3 P.2244

    https://doi.org/10.1002/num.22685 [Citations: 6]

  12. A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models

    Rana, Gule | Asif, Muhammad | Haider, Nadeem | Bilal, Rubi | Ahsan, Muhammad | Al-Mdallal, Qasem | Jarad, Fahd | Napoli, Anna

    Journal of Function Spaces, Vol. 2022 (2022), Iss. P.1

    https://doi.org/10.1155/2022/1541486 [Citations: 5]

  13. A Deep Neural Network Based on ResNet for Predicting Solutions of Poisson–Boltzmann Equation

    Kwon, In | Jo, Gwanghyun | Shin, Kwang-Seong

    Electronics, Vol. 10 (2021), Iss. 21 P.2627

    https://doi.org/10.3390/electronics10212627 [Citations: 2]

  14. A fully decoupled iterative method with three-dimensional anisotropic immersed finite elements for Kaufman-type discharge problems

    Lu, Chang | Wan, Jie | Cao, Yong | He, Xiaoming

    Computer Methods in Applied Mechanics and Engineering, Vol. 372 (2020), Iss. P.113345

    https://doi.org/10.1016/j.cma.2020.113345 [Citations: 12]

  15. A posteriori error estimation and adaptive mesh refinement for parabolic interface problems using non-conforming immersed finite element method

    Ray, Tanushree | Sinha, Rajen Kumar

    Journal of Computational and Applied Mathematics, Vol. 419 (2023), Iss. P.114714

    https://doi.org/10.1016/j.cam.2022.114714 [Citations: 0]

  16. Meshless and multi-resolution collocation techniques for parabolic interface models

    Haider, Nadeem | Aziz, Imran

    Applied Mathematics and Computation, Vol. 335 (2018), Iss. P.313

    https://doi.org/10.1016/j.amc.2018.04.044 [Citations: 3]

  17. An immersed selective discontinuous Galerkin method in particle-in-cell simulation with adaptive Cartesian mesh and polynomial preserving recovery

    Wu, Siyu | Bai, Jinwei | He, Xiaoming | Zhao, Ren | Cao, Yong

    Journal of Computational Physics, Vol. 498 (2024), Iss. P.112703

    https://doi.org/10.1016/j.jcp.2023.112703 [Citations: 0]

  18. Discontinuous bubble immersed finite element method for Poisson-Boltzmann-Nernst-Planck model

    Kwon, In | Kwak, Do Y. | Jo, Gwanghyun

    Journal of Computational Physics, Vol. 438 (2021), Iss. P.110370

    https://doi.org/10.1016/j.jcp.2021.110370 [Citations: 5]

  19. Meshless and Multi-Resolution Collocation Techniques for Steady State Interface Models

    Aziz, Imran | Haider, Nadeem

    International Journal of Computational Methods, Vol. 15 (2018), Iss. 01 P.1750073

    https://doi.org/10.1142/S0219876217500736 [Citations: 4]

  20. A multigrid based finite difference method for solving parabolic interface problem

    Feng, Hongsong | Zhao, Shan

    Electronic Research Archive, Vol. 29 (2021), Iss. 5 P.3141

    https://doi.org/10.3934/era.2021031 [Citations: 3]

  21. Locally Conservative Immersed Finite Element Method for Elliptic Interface Problems

    Jo, Gwanghyun | Kwak, Do Y. | Lee, Young-Ju

    Journal of Scientific Computing, Vol. 87 (2021), Iss. 2

    https://doi.org/10.1007/s10915-021-01476-1 [Citations: 4]

  22. A Posteriori Error Estimator for Finite Element Simulation of Electromagnetic Material Processing

    Jesus O., Garcia C. | Jose R., Alves Z. | Julien, Barlier | Francois, Bay

    IEEE Transactions on Magnetics, Vol. 58 (2022), Iss. 12 P.1

    https://doi.org/10.1109/TMAG.2022.3212597 [Citations: 2]

  23. Three‐dimensional immersed finite‐element method for anisotropic magnetostatic/electrostatic interface problems with nonhomogeneous flux jump

    Lu, Chang | Yang, Zhi | Bai, Jinwei | Cao, Yong | He, Xiaoming

    International Journal for Numerical Methods in Engineering, Vol. 121 (2020), Iss. 10 P.2107

    https://doi.org/10.1002/nme.6301 [Citations: 22]

  24. Geometric multigrid algorithms for elliptic interface problems using structured grids

    Jo, Gwanghyun | Kwak, Do Y.

    Numerical Algorithms, Vol. 81 (2019), Iss. 1 P.211

    https://doi.org/10.1007/s11075-018-0544-9 [Citations: 14]

  25. Weak scaling of the parallel immersed-finite-element particle-in-cell (PIFE-PIC) framework with lunar plasma charging simulations

    Lund, David | He, Xiaoming | Zhang, Xu | Han, Daoru

    Computational Particle Mechanics, Vol. 9 (2022), Iss. 6 P.1279

    https://doi.org/10.1007/s40571-022-00470-0 [Citations: 3]

  26. PIFE-PIC: Parallel Immersed Finite Element Particle-in-Cell for 3-D Kinetic Simulations of Plasma-Material Interactions

    Han, Daoru | He, Xiaoming | Lund, David | Zhang, Xu

    SIAM Journal on Scientific Computing, Vol. 43 (2021), Iss. 3 P.C235

    https://doi.org/10.1137/20M137344X [Citations: 13]

  27. Interpolating stabilized moving least squares (MLS) approximation for 2D elliptic interface problems

    Dehghan, Mehdi | Abbaszadeh, Mostafa

    Computer Methods in Applied Mechanics and Engineering, Vol. 328 (2018), Iss. P.775

    https://doi.org/10.1016/j.cma.2017.09.002 [Citations: 39]

  28. Kinetic Particle Simulations of Plasma Charging at Lunar Craters Under Severe Conditions

    Lund, David | He, Xiaoming | Han, Daoru

    Journal of Spacecraft and Rockets, Vol. 60 (2023), Iss. 4 P.1176

    https://doi.org/10.2514/1.A35622 [Citations: 2]

  29. Fully discrete least-squares spectral element method for parabolic interface problems

    Kishore Kumar, N. | Biswas, Pankaj

    Mathematics and Computers in Simulation, Vol. 181 (2021), Iss. P.364

    https://doi.org/10.1016/j.matcom.2020.10.001 [Citations: 2]

  30. A Huygens immersed-finite-element particle-in-cell method for modeling plasma-surface interactions with moving interface

    Cao, Huijun | Cao, Yong | Chu, Yuchuan | He, Xiaoming | Lin, Tao

    Communications in Nonlinear Science and Numerical Simulation, Vol. 59 (2018), Iss. P.132

    https://doi.org/10.1016/j.cnsns.2017.10.015 [Citations: 16]