Fast multilevel augmentation methods for nonlinear boundary value problems. (English) Zbl 1217.65100
Summary: We develop a multilevel augmentation method for solving nonlinear boundary value problems. We first describe the multilevel augmentation method for solving nonlinear operator equations of the second kind which has been proposed in a recent paper of Z. Chen, S. Cheng and the author [J. Math. Anal. Appl. 375, No. 2, 706–724 (2011; Zbl 1213.35035)], and then apply it to solving the nonlinear two-point boundary value problems of second-order differential equations. The theoretical analysis of convergence order and computational complexity are proposed. Finally numerical experiments are presented to confirm the theoretical estimates and illustrate the efficiency of the method.
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
Keywords:
multilevel augmentation methods; nonlinear operator equations; nonlinear boundary value problemsCitations:
Zbl 1213.35035References:
| [1] | Anne, C.; Bowers, K., The Schwartz alternating sinc domain decomposition method, Appl. Numer. Math., 25, 461-483 (1997) · Zbl 0887.65087 |
| [2] | Burden, R. L.; Faires, J. D., Numerical Analysis (1993), PWS: PWS Boston · Zbl 0788.65001 |
| [3] | Ralph, C.; Bowers, K., The sinc-Galerkin method for fourth-order differential equations, SIAM J. Numer. Anal., 28, 760-788 (1991) · Zbl 0735.65058 |
| [4] | Chawla, M.; Katta, C., Finite difference methods for two-point boundary value problems involving high order differential equations, BIT, 19, 27-33 (1979) · Zbl 0401.65053 |
| [5] | Bialecki, B., Sinc-collocation methods for two-point boundary value problems, IMA J. Numer. Anal., 11, 357-375 (1991) · Zbl 0735.65052 |
| [6] | D. Jespersen, Ritz-Galerkin methods for singular boundary value problems, Math. Research Center Tech. Rep, 1972, Univ of Wisconsin, Madisom, 1977.; D. Jespersen, Ritz-Galerkin methods for singular boundary value problems, Math. Research Center Tech. Rep, 1972, Univ of Wisconsin, Madisom, 1977. · Zbl 0393.65043 |
| [7] | Keller, H. B., (Aziz, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (1975), Academic Press), 27-88 · Zbl 0311.65050 |
| [8] | Chen, Z.; Micchelli, C. A.; Xu, Y., A multilevel method for solving operator equations, J. Math. Anal. Appl., 262, 688-699 (2001) · Zbl 0990.65059 |
| [9] | Chen, Z.; Micchelli, C. A.; Xu, Y., Fast collocation methods for second kind integral equations, SIAM J. Numer. Anal., 40, 344-375 (2002) · Zbl 1016.65107 |
| [10] | Chen, Z.; Wu, B.; Xu, Y., Multilevel augmentation methods for solving operator equations, Numer. Math. J. Chinese Univ., 14, 31-55 (2005) · Zbl 1077.65053 |
| [11] | Chen, Z.; Wu, B.; Xu, Y., Fast multilevel augmentation methods for solving Hammerstein equations, SIAM J. Numer. Anal., 47, 2321-2346 (2009) · Zbl 1195.65224 |
| [12] | Chen, Z.; Xu, Y.; Yang, H., A multilevel augmentation method for solving ill-posed operator equations, Inverse Problems, 22, 155-174 (2006) · Zbl 1088.65051 |
| [13] | Fang, W.; Ma, F.; Xu, Y., Multi-level iteration methods for solving integral equations of the second kind, J. Integral Equations Appl., 14, 355-376 (2002) · Zbl 1041.65107 |
| [14] | Chen, Z.; Wu, B.; Xu, Y., Multilevel augmentation methods for differential equations, Adv. Comput. Math., 24, 213-238 (2006) · Zbl 1123.65075 |
| [15] | Lu, Y.; Shen, L.; Xu, Y., Shadow block iteration for solving linear systems obtained from wavelet transforms, Appl. Comput. Harmon. Anal., 19, 359-385 (2005) · Zbl 1086.65029 |
| [16] | Chen, J.; Chen, Z.; Cheng, S., Multilevel augmentation methods for solving the Sine-Gordon equation, J. Math. Anal. Appl., 375, 706-724 (2011) · Zbl 1213.35035 |
| [17] | Chen, M.; Chen, Z.; Chen, G., Approximate Solutions of Operator Equations (1997), World Scientific Publishing Co.: World Scientific Publishing Co. Singapore · Zbl 0914.65062 |
| [18] | Vainikko, G. M., Perturbed Galerkin method and general theory of approximate methods for nonlinear equations, Zh. Vychisl. Mat. Mat. Fiz., 7, 732-751 (1967), Engl. Translation, USSR Comput. Math. Math. Phys., 7 (1967), 1-41 · Zbl 0213.16201 |
| [19] | Atkinson, K. E.; Potra, F. A., Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal., 24, 1352-1373 (1987) · Zbl 0655.65146 |
| [20] | Ciarlet, P. G.; Schultz, M. H.; Varga, R. S., Numerical methods of high-order accuracy for nonlinear boundary value problems I. One dimensional problems, Numer. Math., 9, 394-430 (1967) · Zbl 0155.20403 |
| [21] | Ciarlet, P. G.; Schultz, M. H.; Varga, R. S., Numerical methods of high-order accuracy for nonlinear boundary value problems IV. Periodic boundary conditions, Numer. Math., 12, 266-279 (1968) · Zbl 0181.18303 |
| [22] | Ciarlet, P. G.; Natterer, F.; Varga, R. S., Numerical methods of high-order accuracy for singular nonlinear boundary value problems, Numer. Math., 15, 87-99 (1970) · Zbl 0211.19103 |
| [23] | Perrin, F. M.; Price, H. S.; Varga, R. S., On high-order numerical methods for nonlinear two-point boundary value problems, Numer. Math., 13, 180-198 (1969) · Zbl 0183.44501 |
| [24] | Berntsen, Jarle; Espelid, Terje O., On the use of gauss quadrature in adaptive automatic integration schemes, BIT, 24, 239-242 (1984) · Zbl 0545.65011 |
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