10.57647/ijm2c.2026.1603.17

Application of lRBF-FD Method for Numerical Solution of the Obstacle Problem

PDF
  • Ahmed Awad Nasser1,
  • Masoud Allame*1,
  • Habeeb Abed Kadhim Aal-Rkhais2,
  • Majid Tavassoli Kajani1,
  1. Department of Mathematics, Isf. C., Islamic Azad University, Isfahan
  2. Department of Mathematics, College of Computer and Mathematics, Uinversity of Thi-Qar, Iraq

Received: 03-11-2025

Revised: 13-12-2025

Accepted: 16-12-2025

Published in Issue 10-02-2026

Published Online: 27-12-2025

Copyright (c) 2025 Ahmed Awad Nasser, Masoud Allame, Habeeb Abed Kadhim Aal-Rkhais, Majid Tavassoli Kajani (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Awad Nasser, A., Allame, M., Abed Kadhim Aal-Rkhais, H., & Tavassoli Kajani, M. (2026). Application of lRBF-FD Method for Numerical Solution of the Obstacle Problem. International Journal of Mathematical Modelling & Computations. https://doi.org/10.57647/ijm2c.2026.1603.17
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

The obstacle problem is a nonlinear variational inequality, first introduced to model contact phenomena in solid mechanics. Because of the singularity of the solution along the free boundary, the development of accurate and efficient numerical methods is critical. This paper introduces a novel meshless technique for numerically solving the obstacle problem, utilizing the local radial basis function-finite difference (lRBF-FD) method. An active set strategy is combined with the lRBF-FD method to manage the complementarity condition of the problem. Notable advantages of this method are its independence from any mesh and its straightforward implementation with high numerical stability. 

Keywords

  • Obstacle problem,
  • Active set method,
  • radial basis functions,
  • meshfree method,
  • Finite difference method
PDF

References

  1. Mostafa Abbaszadeh and Mehdi Dehghan. Meshless upwind local radialbasis fu nction-finite difference technique to simulate the time-fractional distributed-order advection–diffusion equation. Engineering with computers, 37(2):873–889, 2021.
  2. Mostafa Abbaszadeh, Hossein Pourbashash, and Mahmood Khaksar-e Oshagh. The local meshless collocation method for solving 2d fractional 14klein-kramers dynamics equation on irregular domains. International Journal of Numerical Methods for Heat & Fluid Flow, 32(1):41–61, 2022.
  3. Ahmed Awad Nasser, Masoud Allame, Habeeb Abed Kadhim AalRkhais, and Majid Tavassoli Kajani. Application of the meshless interpolating element-free galerkin method for the numerical solution of obstacle problem. Computational Methods for Differential Equations,2025.
  4. Ahmed Awad Nasser, Masoud Allame, Habeeb Abed Kadhim AalRkhais, and Majid Tavassoli-Kajani. Numerical solution of obstacle problem by using the mesh-free radial point interpolation method. Journal of Mathematical Modeling, 13(4):929–942, 2025.
  5. Lothar Banz and Andreas Schr¨oder. Biorthogonal basis functions in hpadaptive fem for elliptic obstacle problems. Computers & Mathematics with Applications, 70(8):1721–1742, 2015.
  6. Victor Bayona, Miguel Moscoso, Manuel Carretero, and Manuel Kindelan. RBF-FD formulas and convergence properties. Journal of Computational Physics, 229(22):8281–8295, 2010.
  7. Victor Bayona, Miguel Moscoso, and Manuel Kindelan. Optimal constant shape parameter for multiquadric based RBF-FD method. Journal of Computational Physics, 230(19):7384–7399, 2011.
  8. Victor Bayona, Miguel Moscoso, and Manuel Kindelan. Gaussian RBFFD weights and its corresponding local truncation errors. Engineering analysis with boundary elements, 36(9):1361–1369, 2012.
  9. Evan F Bollig, Natasha Flyer, and Gordon Erlebacher. Solution to pdes using radial basis function finite-differences (RBF-FD) on multiple GPUs. Journal of Computational Physics, 231(21):7133–7151, 2012.
  10. Erik Burman, Peter Hansbo, Mats G. Larson, and Rolf Stenberg. Galerkin least squares finite element method for the obstacle problem. Computer Methods in Applied Mechanics and Engineering, 313:362 –374, 2017.
  11. Hsin-Fang Chan, Chia-Ming Fan, and Chia-Wen Kuo. Generalized finite difference method for solving two-dimensional non-linear obstacle problems. Engineering Analysis with Boundary Elements, 37(9):1189–1196, 2013.
  12. M Dehghan, M Abbaszadeh, and A Mohebbi. Meshless local PetrovGalerkin and RBFs collocation methods for solving 2D fractional KleinKramers dynamics equation on irregular domains. Comput Model Eng Sci, 107(6):481–516, 2015.
  13. Mehdi Dehghan and Mostafa Abbaszadeh. Proper orthogonal decomposition variational multiscale element free galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation. Computer Methods in Applied Mechanics and Engineering, 311:856–888, 2016.
  14. Gregory E Fasshauer. Meshfree approximation methods with Matlab (With Cd-rom), volume 6. World Scientific Publishing Company, 2007.
  15. Natasha Flyer, Erik Lehto, S´ebastien Blaise, Grady B Wright, and Amik St-Cyr. A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. Journal of Computational Physics, 231(11):4078–4095, 2012.
  16. Bengt Fornberg and Erik Lehto. Stabilization of rbf-generated finite difference methods for convective PDEs. Journal of Computational Physics, 230(6):2270–2285, 2011.
  17. Christian Grossmann, Hans-G¨org Roos, and Martin Stynes. Numerical treatment of partial differential equations. Springer, 2007.
  18. Sara Hosseini. Numerical solution of the parabolic equations by variational iteration method and radial basis functions. International Journal of Mathematical Modelling & Computations, 7(3), 2017.
  19. Volodymyr Hrynkiv. Optimal control of partial differential equations and variational inequalities. The University of Tennessee, 2006.
  20. T K¨arkk¨ainen, K Kunisch, and P Tarvainen. Augmented lagrangian active set methods for obstacle problems. Journal of Optimization Theory and Applications, 119(3):499–533, 2003.
  21. M Khaksar-e Oshagh and Mostafa Shamsi. An adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type. Computers & Mathematics with Applications, 75(2):470–485, 2018.
  22. Mahmood Khaksar-e Oshagh, Mostafa Shamsi, and Mehdi Dehghan. A wavelet-based adaptive mesh refinement method for the obstacle problem. Engineering with Computers, 34:577–589, 2018.
  23. Sajad Kosari and Majid Erfanian. Using chebyshev polynomials zeros as point grid for numerical solution of nonlinear pdes by differential quadrature-based radial basis functions. International Journal of Mathematical Modelling & Computations, 7(01), 2017.
  24. Ruo Li, Wenbin Liu, and Tao Tang. Moving Mesh Finite Element Approximations for Variational Inequality I: Static Obstacle Problem. Hong Kong Baptist University, 2000.
  25. Xiaolin Li and Haiyun Dong. An element-free galerkin method for the obstacle problem. Applied Mathematics Letters, 112:106724, 2021.
  26. Xiao-Peng Lian, Zhongdi Cen, and Xiao-Liang Cheng. Some iterative algorithms for the obstacle problems. International Journal of Computer Mathematics, 87(11):2493–2502, 2010.
  27. Hosein Pourbashash, Khaksar-e Oshagh, Somayyeh Asadollahi, et al. An efficient adaptive wavelet method for pricing time-fractional american option variational inequality. Computational Methods for Differential Equations, 12(1):173–188, 2024.
  28. Hossein Pourbashash and Mahmood Khaksar-e Oshagh. Local RBFFD technique for solving the two-dimensional modified anomalous subdiffusion equation. Applied Mathematics and Computation, 339:144–152, 2018.
  29. MN Rasoulizadeh, O Nikan, and Z Avazzadeh. The impact of LRBFFD on the solutions of the nonlinear regularized long wave equation. Mathematical Sciences, 15(4):365–376, 2021.
  30. Scott A Sarra. A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains. Applied mathematics and Computation, 218(19):9853–9865, 2012.
  31. Varun Shankar, Grady B Wright, Robert M Kirby, and Aaron L Fogelson. A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction–diffusion equations on surfaces. Journal of scientific computing, 63(3):745–768, 2015.
  32. Chang Shu, Hang Ding, and KS Yeo. Local radial basis functionbased differential quadrature method and its application to solve twodimensional incompressible navier–stokes equations. Computer methods in applied mechanics and engineering, 192(7-8):941–954, 2003.
  33. LingDe Su. A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem. Applied Mathematics and Computation, 354:232–247, 2019.
  34. Andrei I Tolstykh. On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations. In Proceedings of the 16th IMACS world congress, volume 228, pages 4606–4624. Lausanne, 2000.
  35. Michael Ulbrich. Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, volume 11. SIAM, 2011.
  36. Holger Wendland. Scattered data approximation, volume 17. Cambridge university press, 2004.
  37. JR Xiao and MA McCarthy. Meshless analysis of the obstacle problem for beams by the mlpg method and subdomain variational formulations. European Journal of Mechanics-A/Solids, 22(3):385–399, 2003.
  38. M Zhao, Huibin Wu, and Chunguang Xiong. Error analysis of hdg approximations for elliptic variational inequality: obstacle problem. Numerical Algorithms, 81:445–463, 2019.

Most read articles by the same author(s)