Article
Full entry |
PDF
(1.2 MB)
Feedback
PDF
(1.2 MB)
Feedback
Keywords:
meshless; element-free Galerkin method; hyperbolic partial differential equation; error estimate; convergence
meshless; element-free Galerkin method; hyperbolic partial differential equation; error estimate; convergence
Summary:
The meshless element-free Galerkin method is developed for numerical analysis of hyperbolic initial-boundary value problems. In this method, only scattered nodes are required in the domain. Computational formulae of the method are analyzed in detail. Error estimates and convergence are also derived theoretically and verified numerically. Numerical examples validate the performance and efficiency of the method.
References:
[1] Abbasbandy, S., Ghehsareh, H. Roohani, Hashim, I., Alsaedi, A.: A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation. Eng. Anal. Bound. Elem. 47 (2014), 10-20. DOI 10.1016/j.enganabound.2014.04.006 | MR 3233886 | Zbl 1297.65125
[2] Belytschko, T., Lu, Y. Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37 (1994), 229-256. DOI 10.1002/nme.1620370205 | MR 1256818 | Zbl 0796.73077
[3] Berger, M. J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984), 484-512. DOI 10.1016/0021-9991(84)90073-1 | MR 0739112 | Zbl 0536.65071
[4] Cheng, Y. M.: Meshless Methods. Science Press, Beijing (2015), Chinese.
[5] Cheng, R.-J., Ge, H.-X.: Element-free Galerkin (EFG) method for a kind of two-dimensional linear hyperbolic equation. Chin. Phys. B. 18 (2009), 4059-4064. DOI 10.1088/1674-1056/18/10/001
[6] Dehghan, M., Ghesmati, A.: Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation. Eng. Anal. Bound. Elem. 34 (2010), 324-336. DOI 10.1016/j.enganabound.2009.10.010 | MR 2585262 | Zbl 1244.65147
[7] Dehghan, M., Ghesmati, A.: Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method. Eng. Anal. Bound. Elem. 34 (2010), 51-59. DOI 10.1016/j.enganabound.2009.07.002 | MR 2559257 | Zbl 1244.65137
[8] Dehghan, M., Salehi, R.: A method based on meshless approach for the numerical solution of the two-space dimensional hyperbolic telegraph equation. Math. Methods Appl. Sci. 35 (2012), 1220-1233. DOI 10.1002/mma.2517 | MR 2945847 | Zbl 1250.35015
[9] Dehghan, M., Shokri, A.: A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions. Numer. Methods Partial Differ. Equations 25 (2009), 494-506. DOI 10.1002/num.20357 | MR 2483780 | Zbl 1159.65084
[10] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19 American Mathematical Society, Providence (2010). DOI 10.1090/gsm/019 | MR 2597943 | Zbl 1194.35001
[11] Hu, X., Huang, P., Feng, X.: A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation. Appl. Math., Praha 61 (2016), 27-45. DOI 10.1007/s10492-016-0120-3 | MR 3455166 | Zbl 06562145
[12] Jiang, Z., Su, L., Jiang, T.: A meshfree method for numerical solution of nonhomogeneous time-dependent problems. Abstr. Appl. Anal. 2014 (2014), Article ID 978310, 11 pages. DOI 10.1155/2014/978310 | MR 3246371
[13] Li, X.: Meshless Galerkin algorithms for boundary integral equations with moving least square approximations. Appl. Numer. Math. 61 (2011), 1237-1256. DOI 10.1016/j.apnum.2011.08.003 | MR 2851120 | Zbl 1232.65160
[14] Li, X.: Error estimates for the moving least-square approximation and the element-free Galerkin method in {$n$}-dimensional spaces. Appl. Numer. Math. 99 (2016), 77-97. DOI 10.1016/j.apnum.2015.07.006 | MR 3413894 | Zbl 1329.65274
[15] Li, X., Li, S.: On the stability of the moving least squares approximation and the element-free Galerkin method. Comput. Math. Appl. 72 (2016), 1515-1531. DOI 10.1016/j.camwa.2016.06.047 | MR 3545373 | Zbl 1361.65090
[16] Li, X., Li, S.: Analysis of the complex moving least squares approximation and the associated element-free Galerkin method. Appl. Math. Model. 47 (2017), 45-62. DOI 10.1016/j.apm.2017.03.019 | MR 3659439
[17] 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. 73 (2016), 21-34. DOI 10.1016/j.enganabound.2016.08.012 | MR 3581428
[18] Li, X., Zhang, S., Wang, Y., Chen, H.: Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations. Comput. Math. Appl. 71 (2016), 1655-1678. DOI 10.1016/j.camwa.2016.03.007 | MR 3481094
[19] Liu, G. R.: Meshfree Methods. Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2010). DOI 10.1201/9781420082104 | MR 2574356 | Zbl 1205.74003
[20] Szekeres, B. J., Izsák, F.: Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems. Appl. Math., Praha 62 (2017), 15-36. DOI 10.21136/AM.2017.0385-15 | MR 3615476 | Zbl 06738479
[21] Tang, Y.-Z., Li, X.-L.: Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems. Chin. Phys. B. 26 (2017), 030203. DOI 10.1088/1674-1056/26/3/030203
[22] Thomas, J. W.: Numerical Partial Differential Equations: Finite Difference Methods. Texts in Applied Mathematics 22 Springer, New York (1995). DOI 10.1007/978-1-4899-7278-1 | MR 1367964 | Zbl 0831.65087
[23] Zhang, S., Li, X.: Boundary augmented Lagrangian method for the Signorini problem. Appl. Math., Praha 61 (2016), 215-231. DOI 10.1007/s10492-016-0129-7 | MR 3470774 | Zbl 06562154
Search
Partner of
