In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.
Citation: Changling Xu, Huilai Li. Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems[J]. Electronic Research Archive, 2023, 31(8): 4818-4842. doi: 10.3934/era.2023247
In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.
| [1] | P. G. Ciarlet, The Finite Element Method for Elliptic Problems, 2$^{nd}$ Amsterdam, North-Holland, 1978. https://doi.org/10.1016/0378-4754(80)90078-6 |
| [2] | J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, 2$^{nd}$ Springer-Verlag, Berlin, 1971. |
| [3] |
M. D. Gunzburger, L. S. Hou, Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim., 34 (1996), 1001–1043. https://doi.org/10.1137/S0363012994262361 doi: 10.1137/S0363012994262361
|
| [4] |
L. S. Hou, J. C. Turner, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), 289–315. https://doi.org/10.1007/s002110050146 doi: 10.1007/s002110050146
|
| [5] |
R. Becker, H. Kapp, R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concept, SIAM J. Control Optim., 39 (2000), 113–132. https://doi.org/10.1137/S0363012999351097 doi: 10.1137/S0363012999351097
|
| [6] |
R. H. W. Hoppe, M. Kieweg, A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems, J. Numer. Math., 17 (2009), 219–244. https://doi.org/10.1515/JNUM.2009.012 doi: 10.1515/JNUM.2009.012
|
| [7] |
H. Li, N. Yan, Recovery type superconvergence and a posteriori error estimate for control problems governed by Stokes equations, J. Comput. Appl. Math., 209 (2007), 187–207. https://doi.org/10.1016/j.cam.2006.10.083 doi: 10.1016/j.cam.2006.10.083
|
| [8] |
R. Li, W. B. Liu, N. Yan, A posteriori error estimates of recovery type for distributed convex optimal control problems, J. Sci. Comput., 33 (2007), 155–182. https://doi.org/10.1007/s10915-007-9147-7 doi: 10.1007/s10915-007-9147-7
|
| [9] |
J. Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal., 29 (1992), 303–319. https://doi.org/10.1137/0729020 doi: 10.1137/0729020
|
| [10] |
J. Xu, A novel two-grid method for semilinear equations, SIAM J. Sci. Comput., 15 (1994), 231–237. https://doi.org/10.1137/0915016 doi: 10.1137/0915016
|
| [11] |
J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759–1777. https://doi.org/10.1137/S0036142992232949 doi: 10.1137/S0036142992232949
|
| [12] |
O. Axelsson, W. Layton, A two-level method for the discretization of nonlinear boundary value problems, SIAM J. Numer. Anal., 33 (1996), 2359–2374. https://doi.org/10.1137/S0036142993247104 doi: 10.1137/S0036142993247104
|
| [13] |
C. N. Dawson, M. F. Wheeler, C. S. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal., 35 (1998), 435–452. https://doi.org/10.1137/S0036142995293493 doi: 10.1137/S0036142995293493
|
| [14] |
J. Xu, A. Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comput., 70 (2001), 17–25. https://doi.org/10.1090/S0025-5718-99-01180-1 doi: 10.1090/S0025-5718-99-01180-1
|
| [15] |
Y. Yang, H. Bi, Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems, SIAM J. Numer. Anal., 49 (2011), 1602–1624. https://doi.org/10.1137/100810241 doi: 10.1137/100810241
|
| [16] |
J. Zhou, Two-grid methods for Maxwell eigenvalue problems, SIAM J. Numer. Anal., 52 (2014), 2027–2047. https://doi.org/10.1137/130919921 doi: 10.1137/130919921
|
| [17] |
Y. Liu, Y. W. Du, H. Li, J. F. Wang, A two-grid finite element approximation for a nonlinear time-fractional cable equation, Nonlinear Dyn., 85 (2016), 2535–2548. https://doi.org/10.1007/s11071-016-2843-9 doi: 10.1007/s11071-016-2843-9
|
| [18] |
H. Brunner, N. Yan, Finite element methods for optimal control problems governed by integral equations and integro-differential equations, Numer. Math., 101 (2005), 1–27. https://doi.org/10.1007/s00211-005-0608-3 doi: 10.1007/s00211-005-0608-3
|
| [19] |
T. Hou, Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro-differential equations, Numer. Methods Partial Differ. Equations, 29 (2013), 1675–1693. https://doi.org/10.1002/num.21771 doi: 10.1002/num.21771
|
| [20] |
C. Meyer, A. Rösch, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), 970–985. https://doi.org/10.1137/S0363012903431608 doi: 10.1137/S0363012903431608
|
| [21] |
Y. P. Lin, V. Thomée, L. B. Wahlbin, Ritz-volterra projections to finite-element spaces and applications to integro-differential and related equations, SIAM J. Numer. Anal., 28 (1991), 1047–1070. https://doi.org/10.1137/0728056 doi: 10.1137/0728056
|
| [22] |
J. Douglas, J. E. Roberts, Global estimates for mixed finite element methods for second order elliptic equations, Math. Comput., 44 (1985), 39–52. https://doi.org/10.1090/S0025-5718-1985-0771029-9 doi: 10.1090/S0025-5718-1985-0771029-9
|
| [23] |
Y. Tang, Y Hua, Superconvergence analysis for parabolic optimal control problems, Calcolo, 51 (2014), 381–392. https://doi.org/10.1007/s10092-013-0091-7 doi: 10.1007/s10092-013-0091-7
|
| [24] |
T. F. Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal., 22 (1985), 970–1013. https://doi.org/10.1137/0722059 doi: 10.1137/0722059
|
| [25] |
O. C. Zienkiwicz, J. Z. Zhu, The supercovergent path recovery and a posteriori error estimates, Part 2: Error estimates and adaptivity, Int. J. Numer. Methods Eng., 33 (1992), 1365–1382. https://doi.org/10.1002/nme.1620330703 doi: 10.1002/nme.1620330703
|
| [26] |
O. C. Zienkiwicz, J. Z. Zhu, The supercovergent path recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Eng., 101 (1992), 207–224. https://doi.org/10.1016/0045-7825(92)90023-D doi: 10.1016/0045-7825(92)90023-D
|
| [27] | L. Chen, An Integrated Finite Element Methods Package in MATLAB, University of California at Irvine, California, 2009. |
Changling Xu, Huilai Li. Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems[J]. Electronic Research Archive, 2023, 31(8): 4818-4842. doi: 10.3934/era.2023247
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