A mixed finite element method for elliptic optimal control problems using a three-field formulation

Abstract

In this paper, we consider an optimal control problem governed by elliptic differential equations posed in a three-field formulation. Using the gradient as a new unknown we write a weak equation for the gradient using a Lagrange multiplier. We use a biorthogonal system to discretise the gradient, which leads to a very efficient numerical scheme. A numerical example is presented to demonstrate the convergence of the finite element approach. References

• D. Boffi, F. Brezzi, and M. Fortin. Mixed finite element methods and applications. Springer–Verlag, 2013. doi:10.1007/978-3-642-36519-5.
• S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer–Verlag, New York, 1994. doi:10.1007/978-0-387-75934-0.
• Yanping Chen. Superconvergence of quadratic optimal control problems by triangular mixed finite element methods. International journal for numerical methods in engineering, 75(8):881–898, 2008. doi:10.1002/nme.2272.
• Hongfei Fu, Hongxing Rui, Jian Hou, and Haihong Li. A stabilized mixed finite element method for elliptic optimal control problems. Journal of Scientific Computing, 66(3):968–986, 2016. doi:10.1007/s10915-015-0050-3.
• Hui Guo, Hongfei Fu, and Jiansong Zhang. A splitting positive definite mixed finite element method for elliptic optimal control problem. Applied Mathematics and Computation, 219(24):11178–11190, August 2013. doi:10.1016/j.amc.2013.05.020.
• Muhammad Ilyas and Bishnu P. Lamichhane. A stabilised mixed finite element method for the poisson problem based on a three-field formulation. In M. Nelson, D. Mallet, B. Pincombe, and J. Bunder, editors, Proceedings of EMAC-2015, volume 57 of ANZIAM J., pages C177–C192. Cambridge University Press, 2016. doi:10.21914/anziamj.v57i0.10356.
• Bishnu P Lamichhane, AT McBride, and BD Reddy. A finite element method for a three-field formulation of linear elasticity based on biorthogonal systems. Computer Methods in Applied Mechanics and Engineering, 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008.
• B.P. Lamichhane. Inf-sup stable finite element pairs based on dual meshes and bases for nearly incompressible elasticity. IMA Journal of Numerical Analysis, 29:404–420, 2009. doi:10.1093/imanum/drn013.
• B.P. Lamichhane. A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems. Journal of Scientific Computing, 46:379–396, 2011. doi:10.1007/s10915-010-9409-7.
• B.P. Lamichhane and E. Stephan. A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numerical Methods for Partial Differential Equations, 28:1336–1353, 2012. doi:10.1002/num.20683.
• Xianbing Luo, Yanping Chen, and Yunqing Huang. Some error estimates of finite volume element approximation for elliptic optimal control problems. International Journal of Numerical Analysis and Modeling, 10(3):697–711, 2013. http://www.math.ualberta.ca/ijnam/Volume-10-2013/No-3-13/2013-03-11.pdf.
• Fredi Troltzsch. On finite element error estimates for optimal control problems with elliptic PDEs. In International Conference on Large-Scale Scientific Computing, pages 40–53. Springer, 2009. doi:10.1007/978-3-642-\(12535-5_4\).
• Fredi Troltzsch. Optimal control of partial differential equations, volume 112. American Mathematical Society, 2010. http://www.ams.org/books/gsm/112/.