Asymptotic Estimate of a Petrov - Galerkin Method for Nonlinear Operator-Differential Equation

In the current paper, we study a Petrov - Galerkin method for a Cauchy problem for an operator-differential equation with a monotone operator in a separable Hilbert space. The existence and the uniqueness of a strong solution of the Cauchy problem are proved. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The minimal requirements to the operators of the equation were demanded, which guaranteed the convergence of the approximate solutions. There were no assumptions of the structure of the operators. Therefore, the method, specified in this paper, can be applied to a wide class of the parabolic equations as well as to the integral-differential equations. The initial boundary value problem for nonlinear parabolic equations of the fourth order on space variables was considered as the application.

Keywords

Cauchy problem; operator-differential equation; Petrov - Galerkin method; orthogonal projection; convergence rate.

References

1. Egorov I.E., Tikhonova I.M. About Convergence Speed of the Stationary Galerkin Method for the Mixed type. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 40 (299), pp. 53-58.
2. Fedotov E.M. Limit Galerkin - Petrov Schemes for a Nonlinear Convection-Diffusion Equation. Differential equations, 2010, vol. 46, no. 7, pp. 1042-1052. DOI: 10.1134/S0012266110070116
3. Petrov G.I. Application of the Galerkin Method to the Problem of the Stability of Viscous Fluid. Journal of Applied Mathematics and Mechanics, 1940, vol. 4, pp. 1-13.
4. Bialecki B., Ganesh M., Mustapha K. A Petrov - Galerkin Method with Quadrature for Elliptic Boundary Value Problems. IMA Journal of Numerical Analysis, 2004, vol. 24, pp. 157-177. DOI: 10.1093/imanum/24.1.157
5. Lin H., Atluri S.N. Meshless Local Petrov - Galerkin (MLPG) Method for Convection-Diffusion Problems. Computer Modeling in Engineering and Sciences, 2000, vol. 1, no. 2, pp. 45-60.
6. Demkowicz L., Oden J.T. An Adaptive Characteristic Petrov - Galerkin Finite Rlement Method for Convection-Dominated Linear and Nonlinear Parabolic Problems in One Space Variable. Journal of Computational Physics, 1986, vol. 67, pp. 188-213. DOI: 10.1016/0021-9991(86)90121-X
7. Demkowicz L., Oden J.T. An Adaptive Characteristic Petrov - Galerkin Finite Element Method for Convection-Dominated Linear and Nonlinear Parabolic Problems Two Space Variable. Computer Methods in Applied Mechanics and Engineering, 1986, vol. 55, pp. 63-87. DOI: 10.1016/0045-7825(86)90086-1
8. Daugavet I.K. On the Method of Moments for Ordinary Differential Equations. Siberian Mathematical Journal, 1965, vol. 6, no. 1, pp. 70-85.
9. Vainikko G.M. Speed of Convergence of the Method of Moments for Ordinary Differential Equations. Siberian Mathematical Journal, 1968, vol. 9, no. 1, pp. 15-20. DOI: 10.1007/BF02196651
10. Dzishkariani A.V. The Galerkin - Petrov Method with Iterations. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 9, pp. 1260-1269.
11. Zarubin A.G. The Method of Moments for a Class of Nonlinear Equations. Siberian Mathematical Journal, 1978, vol. 19, no. 3, pp. 405-412. DOI: 10.1007/BF01875291
12. Zarubin A.G. On the Rate of Convergence of Projection Methods in the Eigenvalue Problems. Computational Mathematics and Mathematical Physics, 1982, vol. 22, no. 6, pp. 26-35. DOI: 10.1016/0041-5553(82)90093-3
13. Zarubin A.G. The Rate of Convergence of Projection Methods in the Eigenvalue Problem for Equations of Special Form. Computational Mathematics and Mathematical Physics, 1985, vol. 25, no. 4, pp. 1-7. DOI: 10.1016/0041-5553(85)90133-8
14. Vinogradova P.V., Zarubin A.G. Error Estimates for the Galerkin Method as Applied to Time-Dependent Equations. Computational Mathematics and Mathematical Physics, 2009, vol. 49, no. 9, pp. 1567-1575. DOI: 10.1134/S0965542509090115
15. Vinogradova P. Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equation. Numerical Functional Analysis and Optimization, 2010, vol. 31, no. 3, pp. 339-365. DOI: 10.1080/01630561003757728
16. Vinogradova P.V. Galerkin Method for a Nonstationary Equation with a Monotone Operator. Differential Equations. 2010, vol. 46, no. 7, pp. 962-972. DOI: 10.1134/S0012266110070049
17. Beckenbach F., Bellman R. Inequalities. Berlin, Springer, 1961. DOI: 10.1007/978-3-642-64971-4
18. Lions J.L., Magenes E. Probl'emes aux Limites non Homog'enes et Applications. V. 1, 2. Paris, Dunod, 1968.
19. Ladyzhenskaya O.A., Ural'tseva N.N. Linear and Quasilinear Equations of Ellipti Type. N.Y., Academic Press, 1968.