Ram Manohar, Rajen Kumar Sinha
Ram Manohar, Rajen Kumar Sinha. ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR FULLY DISCRETE SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS[J]. Journal of Computational Mathematics, 2022, 40(2): 147-176.
[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Elsevier, 2003. [2] H. Amann and P. Quitter, Semilinear parabolic equations involving measures and low regularity data, Trans. Amer. Math. Soc., 356(2004), 1045-1119. [3] I. Babuška and S. Ohnimus, A posteriori error estimation for the semidiscrete finite element method of parabolic partial differential equations, Comput. Meth. Appl. Mech. Engrg., 190(2001), 4691-4712. [4] M. Bieterman and I. Babuska, The finite element method for parabolic equations:(I) a posteriori estimation, Numer. Math., 40(1982), 339-371. [5] M. Bieterman and I. Babuska, The finite element method for parabolic equations:(II) a posteriori estimation and adaptive approach, Numer. Math., 40(1982), 373-406. [6] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, 2008. [7] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978. [8] A. Demlow, O. Lakkis and C. Makridakis, A posteriori error estimates in the maximum norm for parabolic problems, SIAM J. Numer. Anal., 47(2009), 2157-2176. [9] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems 1:A linear model problem, SIAM J. Numer. Anal., 28(1991), 43-77. [10] E.H. Georgoulis, O. Lakkis and J.M. Virtanen, A posteriori error control for discontinuous Galerkin methods for parabolic problems, SIAM J. Numer. Anal., 49(2011), 427-458. [11] M.D. Gunzburger and L.S. Hou, Finite dimensional approximation of a class of constrained nonlinear control problems, SIAM J. Control Optim., 34(1996), 1001-1043. [12] F. Hetch, New development in freeFem++, J. Numer. Math., 20(2012), 251-265. [13] C. Johnson, Y.Y. Nie and V. Thomée, An a posteriori error estimate and adaptive time-step control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal., 27(1990), 277-291. [14] G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20(1982), 414-427. [15] I. Kyza and C. Makridakis, Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations, SIAM J. Numer. Anal., 49(2011), 405-426. [16] O. Lakkis and C. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems, Math. Comp., 75:256(2006), 1627-1658. [17] I. Lasiecka, Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions, SIAM J. Control Optim., 22(1984), 477- 500. [18] L. Li, Z. Lu, W. Zhang, F. Huang and Y. Yang, A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem, J. Inequal. Appl., 138(2018), 1-23. [19] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, SpringerVerlag, 1971. [20] J.L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972. [21] W.B. Liu and N. Yan, A posteriori error estimates for convex boundary control problems, SIAM J. Numer. Anal., 39(2001), 73-99. [22] W.B. Liu and N. Yan, Adaptive Finite Methods for Optimal Control Problems Governed by PDEs, Springer, 2008. [23] Z. Lu and Chongqing, New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element method for general nonlinear parabolic optimal control problems, Appl. Math., 61(2016), 135-163. [24] Z. Lu, H. Liu, C. Hou and L. Cao, New aposteriori error estimates for hp version of finite element methods of nonlinear parabolic optimal control problems, J. Inequal. Appl., 62(2016), 1-17. [25] C. Makridakis and R.H. Nochetto, Elliptic reconstruction and a posteriori error estimates for parabolic problems, SIAM J. Numer. Anal., 41(2003), 1585-1594. [26] P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems:Theory, Algorithms and Applications, Marcell Dekker, 1994. [27] P.A. Nguyen and J.P. Raymond, Control localized on thin structures for semilinear parabolic equations, Stud. Math. and Appl., 31(2002), 591-645. [28] R.H. Nochetto, G. Savare and C. Verdi, A posteriori error estimates for variable time-step discretization of nonlinear evaluation equations, Comm. Pure Appl. Math., 53(2000), 525-589. [29] J.P. Raymond and H. Zidani, Optimal control problem governed by a semilinear parabolic equation, System Modelling and Optimization, 23(1996), 211-217. [30] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54(1990), 483-493. [31] R.K. Sinha and B. Deka, Finite element methods for semilinear elliptic and parabolic interface problems, Appl. Numer. Math., 59(2009), 1870-1883. [32] R.K. Sinha and J. Sen Gupta, A posteriori error analysis for semilinear parabolic interface problem by using elliptic reconstruction, Appl. Anal., 97(2018), 552-570. [33] F. Tröltzsch, Optimal Control of Partial Differential Equations, AMS, Provodence, RI, 2010. [34] R. Verfürth, A posteriori error estimates for nonlinear problems:Lr(0; T; Lρ(Ω))-error estimate for finite element discretization of parabolic equations, Math. Comp., 67(1998), 1335-1360. [35] M.F. Wheeler, A priori L2-error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10(1973), 723-759. [36] F.B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal., 32(1979), 277- 296. |
[1] | Weijie Huang, Wei Jiang, Yan Wang. A θ-L APPROACH FOR SOLVING SOLID-STATE DEWETTING PROBLEMS [J]. Journal of Computational Mathematics, 2022, 40(2): 275-293. |
[2] | Kaibo Hu, Ragnar Winther. WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS [J]. Journal of Computational Mathematics, 2021, 39(3): 333-357. |
[3] | Xiaodi Zhang, Weiying Zheng. MONOLITHIC MULTIGRID FOR REDUCED MAGNETOHYDRODYNAMIC EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(3): 453-470. |
[4] | Xiaoliang Song, Bo Chen, Bo Yu. ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION [J]. Journal of Computational Mathematics, 2021, 39(3): 471-492. |
[5] | Xiaocui Li, Xu You. MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 130-146. |
[6] | Michael Holst, Yuwen Li, Adam Mihalik, Ryan Szypowski. CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK [J]. Journal of Computational Mathematics, 2020, 38(5): 748-767. |
[7] | Qilong Zhai, Xiaozhe Hu, Ran Zhang. THE SHIFTED-INVERSE POWER WEAK GALERKIN METHOD FOR EIGENVALUE PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(4): 606-623. |
[8] | Weifeng Zhang, Shuo Zhang. ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS [J]. Journal of Computational Mathematics, 2020, 38(4): 565-579. |
[9] | Juncai He, Lin Li, Jinchao Xu, Chunyue Zheng. RELU DEEP NEURAL NETWORKS AND LINEAR FINITE ELEMENTS [J]. Journal of Computational Mathematics, 2020, 38(3): 502-527. |
[10] | Jie Chen, Zhengkang He, Shuyu Sun, Shimin Guo, Zhangxin Chen. EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(3): 452-468. |
[11] | Li Cai, Ye Sun, Feifei Jing, Yiqiang Li, Xiaoqin Shen, Yufeng Nie. A FULLY DISCRETE IMPLICIT-EXPLICIT FINITE ELEMENT METHOD FOR SOLVING THE FITZHUGH-NAGUMO MODEL [J]. Journal of Computational Mathematics, 2020, 38(3): 469-486. |
[12] | Huoyuan Duan, Roger C. E. Tan. ERROR ANALYSIS OF A STABILIZED FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES PROBLEM [J]. Journal of Computational Mathematics, 2020, 38(2): 254-290. |
[13] | Carsten Carstensen, Sophie Puttkammer. HOW TO PROVE THE DISCRETE RELIABILITY FOR NONCONFORMING FINITE ELEMENT METHODS [J]. Journal of Computational Mathematics, 2020, 38(1): 142-175. |
[14] | Yu Du, Haijun Wu, Zhimin Zhang. SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS [J]. Journal of Computational Mathematics, 2020, 38(1): 223-238. |
[15] | Weijie Huang, Zhiping Li. A MIXED FINITE ELEMENT METHOD FOR MULTI-CAVITY COMPUTATION IN INCOMPRESSIBLE NONLINEAR ELASTICITY [J]. Journal of Computational Mathematics, 2019, 37(5): 609-628. |
Viewed | ||||||
Full text |
|
|||||
Abstract |
|
|||||