×

High-order upwind finite volume element method for first-order hyperbolic optimal control problems. (English) Zbl 1376.65101

Summary: We present a high-order upwind finite volume element method to solve optimal control problems governed by first-order hyperbolic equations. The method is efficient and easy for implementation. Both the semi-discrete error estimates and the fully discrete error estimates are derived. Optimal order error estimates in the sense of \(L^{2}\)-norm are obtained. Numerical examples are provided to confirm the effectiveness of the method and the theoretical results.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M25 Discrete approximations in optimal control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R. A.; Fournier, J. J. F., Sobolev spaces, pp., (2003), Academic Press, New York · Zbl 1098.46001
[2] Castillo, P.; Cockburn, B.; Perugia, I.; Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38, 1676-1706, (2000) · Zbl 0987.65111 · doi:10.1137/S0036142900371003
[3] Chen, Y.; Yi, N.; Liu, W., A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46, 2254-2275, (2008) · Zbl 1175.49003 · doi:10.1137/070679703
[4] Christofides, P. D., Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes, pp., (2001), Birkhäuser, Basel · Zbl 1018.93001 · doi:10.1007/978-1-4612-0185-4
[5] Hinze, M., A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl., 30, 45-61, (2005) · Zbl 1074.65069 · doi:10.1007/s10589-005-4559-5
[6] Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S., Optimization with PDE constraints, Volume 23 of, pp., (2009), Springer, Netherlands · Zbl 1167.49001
[7] Ito, K.; Kunisch, K., Lagrange multiplier approach to variational problems and applications, Advances in Design and Control, Volume 15, pp., (2008), SIAM, Philadelphia, PA · Zbl 1156.49002
[8] Li, R.; Chen, Z.; Wu, W., Generalized difference methods for differential equations: numerical analysis of finite volume methods, Volume 226, pp., (2000), CRC Press, Florida · Zbl 0940.65125
[9] Lions, J. L., Optimal control of systems governed by partial differential equations, Grundlehren der mathematischen Wissenschaften (trans. S. K. Mitter), Volume 170, pp., (1971), Springer, Berlin-Heidelberg · Zbl 0203.09001
[10] Lions, J. L.; Magenes, E.; Kenneth, P., Non-homogeneous boundary value problems and applications, Volume 1, pp., (1972), Springer, Berlin · Zbl 0227.35001
[11] Luo, X.; Chen, Y.; Huang, Y., A priori error estimates of finite volume element method for hyperbolic optimal control problems, Sci. China Math., 56, 901-914, (2013) · Zbl 1264.65179 · doi:10.1007/s11425-013-4573-5
[12] Meidner, D.; Vexler, B., A priori error estimates for space-time finite element discretization of parabolic optimal control problems. part II: problems with control constraints, SIAM J. Control Optim., 47, 1301-1329, (2008) · Zbl 1161.49035 · doi:10.1137/070694028
[13] Singh, P.; Sharma, K. K., Numerical approximations to the transport equation arising in neuronal variability, Int. J. Pure Appl. Math., 69, 341-356, (2011) · Zbl 1218.92022
[14] Tröltzsch, F., Optimal control of partial differential equations: theory, methods, and applications, Volume 112 of, pp., (2010), American Mathematical Society, Providence, RI · Zbl 1195.49001
[15] Ulbrich, S., A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41, 740-797, (2002) · Zbl 1019.49026 · doi:10.1137/S0363012900370764
[16] Wang, L.; Xu, X., The mathematical foundation of finite element methods (in Chinese), pp., (2004), Science Press, Beijing
[17] Wang, P.; Zhang, Z., Quadratic finite volume element method for the air pollution model, Int. J. Comput. Math., 87, 2925-2944, (2010) · Zbl 1203.92066 · doi:10.1080/00207160802680663
[18] Wang, Q.; Lin, S.; Zhang, Z., Numerical methods for a fluid mixture model, Internat. J. Numer. Methods Fluids, 71, 1-12, (2013) · Zbl 1430.76387 · doi:10.1002/fld.3639
[19] Wheeler, M. F., A priori \(L_2\) error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10, 723-759, (1973) · Zbl 0232.35060 · doi:10.1137/0710062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.