Optimal parameter for the stabilised five-field extended Hu–Washizu formulation

We present a mixed finite element method for the elasticity problem. We expand the standard Hu–Washizu formulation to include a pressure unknown and its Lagrange multiplier. By doing so, we derive a five-field formulation. We apply a biorthogonal system that leads to an efficient numerical formulation. We address the coercivity problem by adding a stabilisation term with a parameter. We also present an analysis of the optimal choices of parameter approximation.\nReferences\n\nI. Babuska and T. Strouboulis. The finite element method and its reliability. Oxford University Press, New York, 2001. https://global.oup.com/academic/product/the-finite-element-method-and-its-reliability-9780198502760?cc=au\u0026lang=en\u0026.\nD. Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge, UK, 3rd edition edition, 2007. doi:10.1017/CBO9780511618635.\nJ. K. Djoko and B. D. Reddy. An extended Hu–Washizu formulation for elasticity. Comput. Meth. Appl. Mech.Eng. 195(44):6330–6346, 2006. doi:10.1016/j.cma.2005.12.013.\nJ. Droniou, M. Ilyas, B. P. Lamichhane, and G. E. Wheeler. A mixed finite element method for a sixth-order elliptic problem. IMA J. Numer. Anal. 39(1):374–397, 2017. doi:10.1093/imanum/drx066.\nM. Ilyas. Finite element methods and multi-field applications. PhD thesis, University of Newcastle, 2019. http://hdl.handle.net/1959.13/1403421.\nM. Ilyas and B. P. Lamichhane. A stabilised mixed finite element method for the Poisson problem based on a three-field formulation. In Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J. pages C177–C192, 2016. doi:10.21914/anziamj.v57i0.10356.\nM. Ilyas and B. P. Lamichhane. A three-field formulation of the Poisson problem with Nitsche approach. In Proceedings of the 13th Biennial Engineering Mathematics and Applications Conference, EMAC-2017, volume 59 of ANZIAM J. pages C128–C142, 2018. doi:10.21914/anziamj.v59i0.12645.\nB. P. Lamichhane. Two simple finite element methods for Reissner–Mindlin plates with clamped boundary condition. Appl. Numer. Math. 72:91–98, 2013. doi:10.1016/j.apnum.2013.04.005.\nB. P. Lamichhane and E. P. Stephan. A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numer. Meth. Part. Diff. Eq. 28(4):1336–1353, 2011. doi:10.1002/num.20683.\nB. P. Lamichhane, A. T. McBride, and B. D. Reddy. A finite element method for a three-field formulation of linear elasticity based on biorthogonal systems. Comput. Meth. Appl. Mech. Eng. 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008.\nJ. C. Simo and F. Armero. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Meth. Eng. 33(7):1413–1449, may 1992. doi:10.1002/nme.1620330705.\nA. Zdunek, W. Rachowicz, and T. Eriksson. A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity. Comput. Math. Appl. 72(1):25–47, 2016. doi:10.1016/j.camwa.2016.04.022.