反射偏微分方程解的惩罚近似
A Penalty Approximation Method for Reflected Partial Differential Equations
摘要: 本文研究了一类二阶非线性抛物型偏微分方程反射问题，通过构造反射项的近似形式，给出了此反射方程的近似方程，即反射偏微分方程的惩罚方程。为了弱化条件，我们引入与反射问题等价的变分不等式问题，证明了惩罚方程的解收敛于变分不等式问题的解，此外由于反射项近似的特殊性，我们得到了其收敛速度，并且可以通过调整惩罚方程中的参数来控制反射方程解的精度。
Abstract: In this article, we mainly study a class of second-order nonlinear parabolic partial differential equation with one reflecting wall. By constructing an approximate form of the reflection term, we provide an approximate equation for this reflection equation, which is the penalty equation for the reflection partial differential equation. In order to weaken the condition, we introduce a variational inequality problem equivalent to the reflection problem and prove that the solution of the penalty equation converges to the solution of the variational inequality problem. Furthermore, due to the particularity of the approximate reflection term, the convergence rate of the solution is given. And the accuracy of the reflection equation solution can be controlled by adjusting the parameters in the penalty equation.
文章引用：马治山, 杜巩胜. 反射偏微分方程解的惩罚近似[J]. 理论数学, 2024, 14(2): 719-732. https://doi.org/10.12677/PM.2024.142071

参考文献

[1] Lions, J.L. and Stampacchia, G. (1967) Variational Inequalities. Communications on Pure and Applied Mathematics, 20, 493-519. [Google Scholar] [CrossRef] 
[2] Karamardian, S. (1971) Generalized Complementarity Problem. Journal of Optimization Theory and Applications, 8, 161-168. [Google Scholar] [CrossRef] 
[3] Huang, Y.S. and Zhou, Y.Y. (2003) Finite-Dimensional Approximation for a Class of Elliptic Obstacle Problems. Nonlinear Analysis: Theory, Methods & Applications, 52, 1745-1754. [Google Scholar] [CrossRef] 
[4] Bergounioux, M. (1997) Use of Augmented Lagrangian Methods for the Optimal Control of Obstacle Problems. Journal of Optimization Theory and Applications, 95, 101-126. [Google Scholar] [CrossRef] 
[5] Wang, S. and Huang, C.-S. (2008) A Power Penalty Method for Solving a Nonlinear Parabolic Complementarity Problem. Nonlinear Analysis: Theory, Methods & Applications, 69, 1125-1137. [Google Scholar] [CrossRef] 
[6] Rodrigues, J.F. (1987) Obstacle Problems in Mathe-matical Physics. North­Holland Publishing, Amsterdam.
[7] Li, W. and Wang, S. (2009) Penalty Approach to the HJB Equation Arising in European Stock Option Pricing with Proportional Transaction Costs. Journal of Optimization Theory and Applications, 143, 279-293. [Google Scholar] [CrossRef] 
[8] Duan, Y., Wu, P. and Zhou, Y. (2023) Penalty Approximation Method for a Double Obstacle Quasilinear Parabolic Variational Inequality Problem. Journal of Industrial and Man-agement Optimization, 19, 1770-1789. [Google Scholar] [CrossRef] 
[9] Wang, S. (2018) An Interior Penalty Method for a Large-Scale Fi-nite-Dimensional Nonlinear Double Obstacle Problem. Applied Mathematical Modelling, 58, 217-228. [Google Scholar] [CrossRef] 
[10] Zhou, Y.Y., Wang, S. and Yang, X.Q. (2014) A Penalty Ap-proximation Method for a Semilinear Parabolic Double Obstacle Problem. Journal of Global Optimization, 60, 531-550. [Google Scholar] [CrossRef] 
[11] Wang, F. and Cheng, X.L. (2008) An Algorithm for Solving the Double Obstacle Problems. Applied Mathematics and Computation, 201, 221-228. [Google Scholar] [CrossRef] 
[12] Konnov, I.V. (2014) Application of the Penalty Method to Nonstationary Approximation of an Optimization Problem. Russian Mathematics, 58, 49-55. [Google Scholar] [CrossRef] 
[13] Kashiwabara, T., Oikawa, I. and Zhou, G. (2016) Penalty Method with P1/P1 Finite Element Approximation for the Stokes Equations under the Slip Boundary Condition. Numerische Mathematik, 134, 705-740. [Google Scholar] [CrossRef] 
[14] Zhou, G., Kashiwabara, T. and Oikawa, I. (2017) A Penalty Method for the Time-Dependent Stokes Problem with the Slip Boundary Condition and Its Finite Element Approximation. Applications of Mathematics, 62, 377-403. [Google Scholar] [CrossRef] 
[15] Zhao, J.X. and Wang, S. (2019) A Power Penalty Approach to a Discretized Obstacle Problem with Nonlinear Constraints. Optimization Letters, 13, 1483-1504. [Google Scholar] [CrossRef] 
[16] Chen, W. and Wang, S. (2014) A Penalty Method for a Fractional Order Parabolic Variational Inequality Governing American Put Option Valuation. Computers & Mathematics with Applications, 67, 77-90. [Google Scholar] [CrossRef] 
[17] Chen, W. and Wang, S. (2017) A Power Penalty Method for a 2D Fractional Partial Differential Linear Complementarity Problem Governing Two-Asset American Option Pricing. Applied Mathematics and Computation, 305, 174-187. [Google Scholar] [CrossRef] 
[18] Duan, Y., Wang, S. and Zhou, Y. (2021) A Power Penalty Ap-proach to a Mixed Quasilinear Elliptic Complementarity Problem. Journal of Global Optimization, 81, 901-918. [Google Scholar] [CrossRef] 
[19] Goeleven, D., Motreanu, D., Dumont, Y. and Rochdi, M. (2003) Variational and Hemivariational Inequalities Theory, Methods, and Applications-Volume I: Unilateral Analysis and Unilateral Mechanics. Springer, New York, 1-205. [Google Scholar] [CrossRef] 

为你推荐