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

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/

A gradient recovery method based on an oblique projection for the virtual element method

The virtual element method is an extension of the finite element method on polygonal meshes. The virtual element basis functions are generally unknown inside an element and suitable projections of the basis functions onto polynomial spaces are used to construct the elemental stiffness and mass matrices. We present a gradient recovery method based on an oblique projection, where the gradient of the L2-polynomial projection of a solution is projected onto a virtual element space. This results in a computationally efficient numerical method. We present numerical results computing the gradients on different polygonal meshes to demonstrate the flexibility of the method. References B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo. Equivalent projectors for virtual element methods. Comput. Math. Appl., 66(3):376–391, 2013. doi:10.1016/j.camwa.2013.05.015. L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. Math. Mod. Meth. Appl. Sci., 23(01): 199–214, 2013. doi:10.1142/S0218202512500492. L. Beirao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. The hitchhiker's guide to the virtual element method. Math. Mod. Meth. Appl. Sci., 24(08): 1541–1573, 2014. doi:10.1142/S021820251440003X. Ilyas, M. and Lamichhane, B. P. and Meylan, M. H. A gradient recovery method based on an oblique projection and boundary modification. In Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2016, volume 58 of ANZIAM J., pages C34–C45, 2017. doi:10.21914/anziamj.v58i0.11730. B. P. Lamichhane. A gradient recovery operator based on an oblique projection. Electron. Trans. Numer. Anal., 37:166–172, 2010. URL http://etna.mcs.kent.edu/volumes/2001-2010/vol37/abstract.php?vol=37&pages=166-172. O. J. Sutton. Virtual element methods. PhD thesis, University of Leicester, Department of Mathematics, 2017. URL http://hdl.handle.net/2381/39955. C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes. Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidiscip. O., 45(3):309–328, 2012. doi:10.1007/s00158-011-0706-z. G. Vacca and L. Beirao da Veiga. Virtual element methods for parabolic problems on polygonal meshes. Numer. Meth. Part. D. E., 31(6): 2110–2134, 2015. doi:10.1002/num.21982. J. Xu and Z. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput., 73:1139–1152, 2004. doi:10.1090/S0025-5718-03-01600-4

Furnace vestibule heat transport models

This is a report on the Lovells Springs challenge that was brought to the Mathematics in Industry Study Group at the University of Newcastle, Australia, in January 2020. The design of a furnace that heats steel rods to make them malleable and allow the reshaping of the rods into coiled springs is the challenge.  Mathematical modelling of heat transport in the half-metre long furnace vestibule predicts the effect of vestibule geometry on the temperature of rods entering the furnace, and provides guidelines for deciding on the dimensions of the vestibule for improved energy efficiency of heating. Models considered include treating the rods as equivalent steel sheets, and as discrete steel rods. The relative importance of radiative and convective heat transfer mechanisms is considered. A longer vestibule, with length one or two metres,  is recommended for improved heating efficiency of rods thicker than 25mm