Error estimates of triangular mixed finite element methods for quasilinear optimal control problems

Abstract

The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.

Keywords

A priori error estimate / quasilinear elliptic equation / general convex optimal control problem / triangular mixed finite element method

References

[1] Arada N., Casas E., Tröltzsch F. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput Optim Appl, 2002, 23(2): 201-229

[2] Babuska I., Strouboulis T. The Finite Element Method and Its Reliability, 2001, Oxford: Oxford University Press

[3] Becker R., Kapp H., Rannacher R. Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J Control Optim, 2000, 39(1): 113-132

[4] Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal Numer, 1974, 8(2): 129-151

[5] Brezzi F., Fortin M. Mixed and Hybrid Finite Element Methods, 1991, Berlin: Springer

[6] Chen Y., Liu W. Posteriori error estimates for mixed finite elements of a quadratic optimal control problem. Recent Progress Comp Appl PDEs, 2002, 3: 123-134

[7] Chen Y., Liu W. Error estimates and superconvergence of mixed finite element for quadratic optimal control. Int J Num Anal Model, 2006, 3(3): 311-321

[8] Chen Y., Liu W. A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J Comput Appl Math, 2008, 211(1): 76-89

[9] Chen Y., Lu Z. Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems. Comput Methods Appl Mech Engrg, 2010, 199(1): 1415-1423

[10] Chen Y., Lu Z. Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Finite Elem Anal Des, 2010, 46(4): 957-965

[11] Falk F. Approximation of a class of optimal control problems with order of convergence estimates. J Math Anal Appl, 1973, 44(1): 28-47

[12] Geveci T. On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO: Numer Anal, 1979, 33: 313-328

[13] Grisvard P. Elliptic Problems in Nonsmooth Domains, 1985, London: Pitman

[14] Gunzburger M., Hou S. Finite dimensional approximation of a class of constrained nonlinear control problems. SIAM J Control Optim, 1996, 34(3): 1001-1043

[15] Li R, Liu W. http://circus.math.pku.edu.cn/AFEPack

[16] Li R., Liu W., Ma H., Tang T. Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J Control Optim, 2002, 41(5): 1321-1349

[17] Lions J. Optimal Control of systems Governed by Partial Differential Equations, 1971, Berlin: Springer

[18] Liu W., Tiba D. Error estimates for the finite element approximation of a class of nonlinear optimal control problems. J Numer Func Optim, 2001, 22: 935-972

[19] Liu W., Yan N. A posteriori error estimates for distributed convex optimal control problems. Adv Comput Math, 2001, 15(4): 285-309

[20] Liu W., Yan N. A posteriori error estimates for control problems governed by nonlinear elliptic equation. Appl Numer Math, 2003, 47(2): 173-187

[21] Lu Z., Chen Y. A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems. Adv Appl Math Mech, 2009, 1(2): 242-256

[22] Lu Z., Chen Y. L^∞-error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation. Numer Anal Appl, 2009, 12(1): 74-86

[23] Lu Z., Chen Y., Zhang H. A priori error analysis of mixed methods for nonlinear quadratic optimal control problems. Lobachevskii J Math, 2008, 29(2): 164-174

[24] Malanowski K. Convergence of approximation vs. regularity of solutions for convex control constrained optimal control systems. Appl Math Optim, 1981, 8(1): 69-95

[25] Miliner F. Mixed finite element methods for quasilinear second-order elliptic problems. Math Comp, 1985, 44(170): 303-320

[26] Raviart P., Thomas J. A mixed finite element method for 2nd order elliptic problems. Math Aspects of the Finite Element Method, 1977, Berlin: Springer, 292-315