The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis

References

• Xu J . A novel two-grid method for semilinear elliptic equations. SIAM J Sci Comput. 1994;15:231–237.
   Web of Science ®Google Scholar
• Liu W . Zhe Yin. J. Li, A two-grid algorithm based on expanded mixed element discretizations for strongly nonlinear elliptic equations, Numer Algor. 2015;70:93–111.
   Google Scholar
• Dai X , Cheng X . A two-grid method based on Newton iteration for the Navier-Stokes equations. J Comput Appl Math. 2008;220:566–573.
   Web of Science ®Google Scholar
• Vulkov LG , Zadorin AI . Two-grid algorithms for an ordinary second-order equation with an exponential boundary layer in the solution. Int J Numer Ana Modell. 2010;7:580–592.
   Web of Science ®Google Scholar
• Caboussat A , Glowinski R . A two-grids/projection algorithm for obstacle problems. Comput Math Appl. 2005;50:171–178.
   Web of Science ®Google Scholar
• Moghaderi H , Dehghan M , Hajarian M . A fast and efficient two-grid method for solving d-dimensional poisson equations. Numer Algor. 2016;72:483–537.
   Web of Science ®Google Scholar
• Bajpai S , Nataraj N . On a two-grid finite element scheme combined with Crank-Nicolson method for the equations of motion arising in the Kelvin-Voigt model. Comput Math Appl. 2014;68:2277–2291.
   Web of Science ®Google Scholar
• Bajpai S , Nataraj N , Pani AK . On a two-grid finite element scheme for the equations of motion arising in Kelvin-Voigt model. Adv Comput Math. 2014;40:1043–1071.
   Web of Science ®Google Scholar
• Bajpai S , Pani AK . On a three level two-grid finite element method for the 2D-transient Navier-Stokes equations. 2014. Preprint. arXiv:1401.5540.
   Google Scholar
• Chunk SK , Pani AK . Numerical methods for the Rosenau equation. Appl Anal. 2001;77:351–369.
   Google Scholar
• Atouani N , Omrani K . Galerkin finite element method for the Rosenau-RLW equation. Comput Math Appl. 2013;66:289–303.
   Web of Science ®Google Scholar
• Chung SK , Ha SN . Finite element Galerkin solutions for the Rosenau equation. Appl Anal. 1994;54:39–56.
   Google Scholar
• Chung SK . Finite difference approximate solutions for the Rosenau equation. Appl Anal. 1998;69(1–2):149–156.
   Google Scholar
• He D , Pan K . A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation. Appl Math Comput. 2015;271:323–336.
   Web of Science ®Google Scholar
• Atouani N , Omrani K . On the convergence of conservative difference schemes for the 2D generalized Rosenau-Korteweg de Vries equation. Appl Math Comput. 2015;250:832–847.
   Web of Science ®Google Scholar
• Atouani N , Omrani K . A new conservative high-order accurate difference scheme for the Rosenau equation. Appl Anal. 2015;94:2435–2455.
   Web of Science ®Google Scholar
• Ghiloufi A , Kadri T . Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation. 2015, doi:10.1080/00036811.2016.1186270.
   Google Scholar
• Pan X , Zhang L . On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation. Appl Math Model. 2012;36:3371–3378.
   Web of Science ®Google Scholar
• Pan X , Zheng K , Zhang L . Finite difference discretization of the Rosenau-RLW equation. Appl Anal. 2013;92:2578–2589.
   Web of Science ®Google Scholar
• Hu JS , Zheng KL . Two conservative difference schemes for the generalized Rosenau equation. Bound Value Prob. 2010;2010: Article ID 543503.
   Google Scholar
• Zuo JM , Zhang YM , Zhang TD . A new conservative difference scheme for the general Rosenau-RLW equation. Bound Value Probl. 2010;2010. Article ID 516260, 13p.
   Web of Science ®Google Scholar
• Shokri A , Dehghan M . A meshless method using the radial basis functions for numerical solution of the regularized long wave equation. Numer Methods Partial Differ. 2010;26:807–825.
   Web of Science ®Google Scholar
• Dehghan M , Salehi R . The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas. Comput Phys Commun. 2011;182:2540–2549.
   Web of Science ®Google Scholar
• Dehghan M , Abbaszadeh M , Mohebbi A . The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J Comput Appl Math. 2015;286:211–231.
   Web of Science ®Google Scholar
• Cai W , Sun Y , Wang Y . Variational discretizations for the generalized Rosenau-type equations. Appl Math Comput. 2015;271:860–873.
   Web of Science ®Google Scholar
• Wazwaz AM . Partial differential equations and solitary waves theory. Beijin: Higher Education Press; 2009.
   Google Scholar
• Belytschko T , Lu YY , Gu L . Element free Galerkin methods. Int J Numer Methods Eng. 1994;37:229–256.
   Web of Science ®Google Scholar
• Belytschko T , Krongauz Y , Organ D , et al . Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng. 1996;139:3–47.
   Web of Science ®Google Scholar
• Chung HJ , Belytschko T . An error estimate in the EFG method. Comput Mech. 1998;21:91–100.
   Web of Science ®Google Scholar
• Lancaster P , Salkauskas K . Surfaces generated by moving least squares methods. Math Comput. 1981;37:141–158.
   Web of Science ®Google Scholar
• Cheng YM , Bai FN , Peng MJ . A novel interpolating element free Galerkin (IEFG) method for two-dimensional elastoplasticity. Appl Math Model. 2014;38:5187–5197.
   Web of Science ®Google Scholar
• Cheng YM , Li J . A meshless method with complex variables for elasticity. Acta Phys Sinic. 2005;54:4463–4471.
   Web of Science ®Google Scholar
• Cheng YM , Peng M . Boundary element free method for elastodynamics. Sci China G. 2005;48:641–657.
   Web of Science ®Google Scholar
• Zhang Z , Liew KM , Cheng Y , Lee YY . Analyzing 2D fracture problems with the improved element free Galerkin method. Eng Anal Bound Elem. 2008;32:241–250.
   Web of Science ®Google Scholar
• Zhang LW , Deng YJ , Liew KM , et al . The improved complex variable element free Galerkin method for two-dimensional Schrödinger equation. Comput Math Appl. 2014;68(10):1093–1106.
   Web of Science ®Google Scholar
• Zhang LW , Deng YJ , Liew KM . An improved element-free Galerkin method for numerical modeling of the biological population problems. Eng Anal Bound Elem. 2014;40:181–188.
   Web of Science ®Google Scholar
• Zhang Z , Hao SY , Liew KM , Cheng YM . The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng Anal Bound Elem. 2013;37:1576–1584.
   Web of Science ®Google Scholar
• Zhang Z , Liew KM , Cheng Y . Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng Anal Bound Elem. 2008;32:100–107.
   Web of Science ®Google Scholar
• Dehghan M , Abbaszadeh M . Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition. Appl Numer Math. 2016;109:208–234.
   Web of Science ®Google Scholar
• Li X . Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces. Appl Numer Math. 2016;99:77–97.
   Web of Science ®Google Scholar
• Li X . Meshless Galerkin algorithms for boundary integral equations with moving least square approximations. Appl Numer Math. 2011;61(12):1237–1256.
   Web of Science ®Google Scholar
• Li X , Chen H , Wang Y . Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method. Appl Math Comput. 2015;262:56–78.
   Web of Science ®Google Scholar
• Li X , Zhang S . Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation. Appl Math Model. 2016;40:2875–2896.
   Web of Science ®Google Scholar
• Li X . An interpolating boundary element-free method for three-dimensional potential problems. Appl Math Model. 2015;39:3116–3134.
   Web of Science ®Google Scholar
• Li X , Wang Q . Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases. Eng Anal Bound Elem. 2016;73:21–34.
   Web of Science ®Google Scholar
• Li X , Li S . On the stability of the moving least squares approximation and the element-free Galerkin method. Comput Math Appl. 2016;72:1515–1531.
   Web of Science ®Google Scholar
• Dehghan M , Abbaszadeh M , Mohebbi A . Error estimate for the numerical solution of fractional reaction-subdiffusion process based on a meshless method. J Comput Appl Math. 2015;280:14–36.
   Web of Science ®Google Scholar
• Dehghan M , Shokri A . A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math Comput Simul. 2008;79:700–715.
   Web of Science ®Google Scholar
• Bui TQ , Zhang C . Moving Kriging interpolation-based meshfree method for dynamic analysis of structures. Proc Appl Math Mech. 2011;11:197–198.
   Google Scholar
• Bui TQ , Nguyen MN , Zhang C . A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis. Comput Methods Appl Mech Eng. 2011;200:1354–1366.
   Web of Science ®Google Scholar
• Chen L , Liew KM . A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems. Comput Mech. 2011;47:455–467.
   Web of Science ®Google Scholar
• Dai BD , Cheng J , Zheng B . Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method. Appl Math Comput. 2013;219:10044–10052.
   Web of Science ®Google Scholar
• Dai BD , Cheng J , Zheng B . A moving Kriging interpolation-based meshless local Petrov-Galerkin method for elastodynamic analysis. Int J Appl Mech. 2013;5(1):1350011–1350021.
   Web of Science ®Google Scholar
• Li XG , Dai BD , Wang LH . A moving Kriging interpolation-based boundary node method for two-dimensional potential problems. Chin Phys B. 2010;19(12):120202–120207.
   Web of Science ®Google Scholar
• Dai KY , Liu GR , Lim KM , Gu YT . Comparison between the radial point interpolation and the Kriging interpolation used in meshfree methods. Comput Mech. 2003;32:60–70.
   Web of Science ®Google Scholar
• Gu L . Moving Kriging interpolation and element-free Galerkin method. Int J Numer Meth Eng. 2003;56:1–11.
   Web of Science ®Google Scholar
• Gu YT , Wang QX , Lam KY . A meshless local Kriging method for large deformation analyses. Comput Methods Appl Mech Eng. 2007;196:1673–1684.
   Web of Science ®Google Scholar
• Gu YT , Wang W , Zhang LC , Feng XQ . An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields. Eng Frac Mech. 2011;78:175–190.
   Web of Science ®Google Scholar
• Lam KY , Wang QX , Li H . A novel meshless approach Local Kriging (LoKriging) method with two-dimensional structural analysis. Comput Mech. 2004;33:235–244.
   Web of Science ®Google Scholar
• Li H , Wang QX , Lam KY . Development of a novel meshless local Kriging (LoKriging) method for structural dynamic analysis. Comput Methods Appl Mech Eng. 2004;193:2599–2619.
   Web of Science ®Google Scholar
• Zheng B , Dai BD . A meshless local moving Kriging method for two-dimensional solids. Appl Math Comput. 2011;218:563–573.
   Web of Science ®Google Scholar
• Zhu P , Zhang LW , Liew KM . Geometrically nonlinear thermo-mechanical analysis of moderately thick functionally graded plates using a local Petrov-Galerkin approach with moving Kriging interpolation. Compos Struct. 2014;107:298–314.
   Web of Science ®Google Scholar
• Gu YT , Liu GR . A local point interpolation method for static and dynamic analysis of thin beams. Comput Methods Appl Mech Eng. 2001;190:5515–5528.
   Web of Science ®Google Scholar
• Gu YT , Liu GR . A boundary point interpolation method for stress analysis of solids. Comput Mech. 2002;28:47–54.
   Web of Science ®Google Scholar
• Tongsuk P , Kanok-Nukulchai W . Further investigation of element free Galerkin method using moving Kriging interpolation. Int J Comput Methods. 2004;1:345–365.
   Web of Science ®Google Scholar
• Brezis H . Functional analysis. Sobolev spaces and partial differential equations. New York (NY): Springer; 2011.
   Google Scholar
• Quarteroni A , Valli A . Numerical approximation of partial differential equations. New York (NY): Springer-Verlag; 1997.
   Google Scholar
• Dehghan M . Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul. 2006;71:16–30.
   Web of Science ®Google Scholar
• Mittal RC , Jain RK . Numerical solution of general Rosenau-RLW equation using quintic B-splines collocation method. Commun Numer Anal. 2012;2012: Article ID cna-00129, 9p.
   Google Scholar