Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method. (English) Zbl 1377.65158
Summary: The main purpose of this article is to investigate a computational scheme for solving a class of nonlinear boundary integral equations which occurs as a reformulation of boundary value problems of Laplace’s equations with nonlinear Robin boundary conditions. The method approximates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals appeared in the method. The numerical scheme developed in the current paper utilizes the non-uniform Gauss-Legendre quadrature rule to estimate logarithm-like singular integrals. Since the proposed method is constructed on a set of scattered points, it does not require any background mesh and so we can call it as the meshless local discrete Galerkin (MLDG) method. The scheme is simple and effective to solve boundary integral equations and its algorithm can be easily implemented. We also obtain the error bound and the convergence rate of the presented method. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.
MSC:
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
nonlinear boundary integral equation; Laplace’s equation; discrete Galerkin method; moving least squares method; meshless method; error analysis; nonlinear Robin boundary conditions; Gauss-Legendre quadrature rule; convergence; algorithm; numerical exampleReferences:
| [1] | Armentano, M. G., Error estimates in Sobolev spaces for moving least square approximations, SIAM J. Numer. Anal., 39, 1, 38-51 (2002) · Zbl 1001.65014 |
| [2] | Armentano, M. G.; Duron, R. G., Error estimates for moving least square approximations, Appl. Numer. Math., 37, 3, 397-416 (2001) · Zbl 0984.65096 |
| [3] | Assari, P.; Dehghan, M., A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions, Eur. Phys. J. Plus, 132, 1-23 (2017) |
| [4] | Assari, P.; Dehghan, M., The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision, Eng. Comput., 33, 4, 853-870 (2017) |
| [5] | Assari, P.; Adibi, H.; Dehghan, M., A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis, J. Comput. Appl. Math., 239, 1, 72-92 (2013) · Zbl 1255.65233 |
| [6] | Assari, P.; Adibi, H.; Dehghan, M., A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method, Appl. Math. Model., 37, 22, 9269-9294 (2013) · Zbl 1427.65415 |
| [7] | Assari, P.; Adibi, H.; Dehghan, M., A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels, J. Comput. Appl. Math., 267, 160-181 (2014) · Zbl 1293.65166 |
| [8] | Assari, P.; Adibi, H.; Dehghan, M., A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains, Numer. Algorithms, 67, 2, 423-455 (2014) · Zbl 1311.65163 |
| [9] | Assari, P.; Adibi, H.; Dehghan, M., The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis, Appl. Numer. Math., 81, 76-93 (2014) · Zbl 1291.65375 |
| [10] | Atkinson, K. E., The numerical solution of a non-linear boundary integral equation on smooth surfaces, IMA J. Numer. Anal., 14, 4, 461-483 (1994) · Zbl 0814.65110 |
| [11] | Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0155.47404 |
| [12] | Atkinson, K. E.; Bogomolny, A., The discrete Galerkin method for integral equations, Math. Comput., 48, 178, 595-616 (1987) · Zbl 0633.65134 |
| [13] | Atkinson, K. E.; Chandler, G., Boundary integral equation methods for solving Laplace’s equation with nonlinear boundary conditions: the smooth boundary case, Math. Comput., 55, 192, 451-472 (1990) · Zbl 0709.65088 |
| [14] | Atkinson, K. E.; Chien, D., Piecewise polynomial collocation for boundary integral equations, SIAM J. Sci. Comput., 16, 3, 651-681 (1995) · Zbl 0826.65095 |
| [15] | Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 2, 117-127 (1998) · Zbl 0932.76067 |
| [16] | Bardhan, J. P., Numerical solution of boundary-integral equations for molecular electrostatics, J. Chem. Phys., 130, 9, 1-13 (2009) |
| [17] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Eng., 37, 2, 229-256 (1994) · Zbl 0796.73077 |
| [18] | Buhmann, M. D., Radial Basis Functions: Theory and Implementations (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1038.41001 |
| [19] | Chen, W.; Lin, W., Galerkin trigonometric wavelet methods for the natural boundary integral equations, Appl. Math. Comput., 121, 1, 75-92 (2001) · Zbl 1026.65111 |
| [20] | Cheng, P.; Huang, J., Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods, Numer. Algorithms, 58, 4, 545-554 (2011) · Zbl 1241.65104 |
| [21] | Cheng, P.; Huang, J.; Wang, Z., Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations, Appl. Math. Mech., 32, 12, 1505-1514 (2011) · Zbl 1238.35016 |
| [22] | Cuomo, S.; Galletti, A.; Giunta, G.; Marcellino, L., Piecewise Hermite interpolation via barycentric coordinates, Memory of Prof. Carlo Ciliberto. Memory of Prof. Carlo Ciliberto, Ric. Mat., 64, 303-319 (2015) · Zbl 1326.41003 |
| [23] | Cuomo, S.; Galletti, A.; Giunta, G.; Marcellino, L., Reconstruction of implicit curves and surfaces via RBF interpolation, Appl. Numer. Math. (2015) |
| [24] | Dehghan, M.; Mirzaei, D., Numerical solution to the unsteady two-dimensional Schrodinger equation using meshless local boundary integral equation method, Int. J. Numer. Methods Eng., 76, 4, 501-520 (2008) · Zbl 1195.81007 |
| [25] | Dehghan, M.; Salehi, R., The numerical solution of the non-linear integro-differential equations based on the meshless method, J. Comput. Appl. Math., 236, 9, 2367-2377 (2012) · Zbl 1243.65154 |
| [26] | Fang, W.; Wang, Y.; Xu, Y., An implementation of fast wavelet Galerkin methods for integral equations of the second kind, J. Sci. Comput., 20, 2, 277-302 (2004) · Zbl 1047.65112 |
| [27] | Fasshauer, G. E., Meshfree methods, (Handbook of Theoretical and Computational Nanotechnology (2005), American Scientific Publishers) |
| [28] | Ganesh, M.; Steinbach, O., Nonlinear boundary integral equations for harmonic problems, J. Integral Equ. Appl., 11, 4, 437-459 (1999) · Zbl 0974.65112 |
| [29] | Gao, J.; Jiang, Y., Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel, J. Comput. Appl. Math., 215, 1, 242-259 (2008) · Zbl 1350.65144 |
| [30] | Golbabai, A.; Seifollahi, S., Numerical solution of the second kind integral equations using radial basis function networks, Appl. Math. Comput., 174, 2, 877-883 (2006) · Zbl 1090.65149 |
| [31] | Harbrecht, H.; Schneider, R., Wavelet Galerkin schemes for boundary integral equations implementation and quadrature, SIAM J. Sci. Comput., 27, 4, 1347-1370 (2006) · Zbl 1117.65162 |
| [32] | Helsing, J.; Karlsson, A., An explicit kernel-split panel-based Nystrom scheme for integral equations on axially symmetric surfaces, J. Comput. Phys., 272, 686-703 (2014) · Zbl 1349.65709 |
| [33] | Kaneko, H.; Xu, Y., Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind, Math. Comput., 62, 206, 739-753 (1994) · Zbl 0799.65023 |
| [34] | Kaneko, H.; Xu, Y., Superconvergence of the iterated Galerkin methods for Hammerstein equations, SIAM J. Numer. Anal., 33, 3, 1048-1064 (1996) · Zbl 0860.65138 |
| [35] | Kress, B., Linear Integral Equations (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0671.45001 |
| [36] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. Comput., 37, 155, 141-158 (1981) · Zbl 0469.41005 |
| [37] | Li, X., Meshless Galerkin algorithms for boundary integral equations with moving least square approximations, Appl. Numer. Math., 61, 12, 1237-1256 (2011) · Zbl 1232.65160 |
| [38] | Li, X., Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces, Appl. Numer. Math., 99, 77-97 (2016) · Zbl 1329.65274 |
| [39] | Li, X.; Li, S., On the stability of the moving least squares approximation and the element-free Galerkin method, Comput. Math. Appl., 72, 1515-1531 (2016) · Zbl 1361.65090 |
| [40] | Li, X.; Li, S., Analysis of the complex moving least squares approximation and the associated element-free Galerkin method, Appl. Math. Model., 47, 45-62 (2017) · Zbl 1446.65176 |
| [41] | 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, 21-34 (2016) · Zbl 1403.65206 |
| [42] | 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., 40, 4, 2875-2896 (2016) · Zbl 1452.65383 |
| [43] | Li, X.; Zhu, J., A Galerkin boundary node method and its convergence analysis, J. Comput. Appl. Math., 230, 1, 314-328 (2009) · Zbl 1189.65291 |
| [44] | Li, X.; Zhu, J., A Galerkin boundary node method for biharmonic problems, Eng. Anal. Bound. Elem., 33, 6, 858-865 (2009) · Zbl 1244.65175 |
| [45] | Li, X.; Zhu, J., A meshless Galerkin method for Stokes problems using boundary integral equations, Comput. Methods Appl. Mech. Eng., 198, 2874-2885 (2009) · Zbl 1229.76076 |
| [46] | 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., 262, 56-78 (2015) · Zbl 1410.65456 |
| [47] | McLean, W., A spectral Galerkin method for a boundary integral equation, Math. Comput., 47, 176, 597-607 (1986) · Zbl 0614.65124 |
| [48] | Mirzaei, D.; Dehghan, M., Implementation of meshless LBIE method to the 2D non-linear SG problem, Int. J. Numer. Methods Eng., 79, 13, 1662-1682 (2009) · Zbl 1176.65118 |
| [49] | Mirzaei, D.; Dehghan, M., A meshless based method for solution of integral equations, Appl. Numer. Math., 60, 3, 245-262 (2010) · Zbl 1202.65174 |
| [50] | Parand, K.; Rad, J. A., Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions, Appl. Math. Comput., 218, 9, 5292-5309 (2012) · Zbl 1244.65245 |
| [51] | Rokhlin, V.; Bremer, J.; Sammis, I., Universal quadratures for boundary integral equations on two-dimensional domains with corners, J. Comput. Phys., 229, 8259-8280 (2010) · Zbl 1201.65213 |
| [52] | Ruotsalainen, K.; Saranen, J., On the collocation method for a nonlinear boundary integral equation, J. Comput. Appl. Math., 28, C, 339-348 (1989) · Zbl 0684.65098 |
| [53] | Salehi, R.; Dehghan, M., A moving least square reproducing polynomial meshless method, Appl. Numer. Math., 69, 34-58 (2013) · Zbl 1284.65137 |
| [54] | Sladek, J.; Sladek, V.; Atluri, S. N., Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties, Comput. Mech., 24, 6, 456-462 (2000) · Zbl 0961.74073 |
| [55] | Tang, X.; Pang, Z.; Zhu, T.; Liu, J., Wavelet numerical solutions for weakly singular Fredholm integral equations of the second kind, Wuhan Univ. J. Nat. Sci., 12, 3, 437-441 (2007) · Zbl 1174.65559 |
| [56] | Von Petersdorff, T.; Schwab, C., Wavelet approximations for first kind boundary integral equations on polygons, Numer. Math., 74, 4, 479-519 (1996) · Zbl 0863.65074 |
| [57] | Wendland, H., Scattered Data Approximation (2005), Cambridge University Press: Cambridge University Press New York · Zbl 1075.65021 |
| [58] | Xie Mukherjee, Y.; Mukherjee, S., The boundary node method for potential problems, Int. J. Numer. Methods Eng., 40, 5, 797-815 (1997) · Zbl 0885.65124 |
| [59] | Yan, Yi, Fast numerical solution for a second kind boundary integral equation with a logarithmic kernel, SIAM J. Numer. Anal., 31, 2, 477-498 (1994) · Zbl 0805.65110 |
| [60] | Zhang, P.; Zhang, Y., Wavelet method for boundary integral equations, J. Comput. Math., 18, 1, 25-42 (2000) · Zbl 0952.65095 |
| [61] | Zuppa, C., Good quality point sets and error estimates for moving least square approximations, Appl. Numer. Math., 47, 3-4, 575-585 (2003) · Zbl 1040.65034 |
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