We present a mixed finite element method for a three-field formulation of the Poisson problem and apply a biorthogonal system leading to an efficient numerical computation. The three-field formulation is similar to the Hu-Washizu formulation for the linear elasticity problem. A parameterised approach is given to stabilise the problem so that its associated bilinear form is coercive on the whole space. Analysis of optimal choices of parameter approximation and numerical examples are provided to evaluate our stabilised form. References P. B. Bochev and C. R. Dohrmann. A computational study of stabilized, low-order \(C^0\) finite element approximations of Darcy equations. Comput. Mech. 38(4):323–333, 2006. doi:10.1007/s00466-006-0036-y D. Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, 3rd edition, 2007. doi:10.1017/CBO9780511618635 S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer, New York, 3rd edition edition, 2008. doi:10.1007/978-0-387-75934-0 B. 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 B. 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. Method. Appl. Mech. Eng. 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008 B. P. Lamichhane and E. P. Stephan. A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numer. Meth. Part. D. E. 28(4):1336–1353, 2012. doi:10.1002/num.20683 S. Micheletti and R. Sacco. Dual-primal mixed finite elements for elliptic problems. Comput. Method. Appl. Mech. Eng. 17(2):137–151, 2001. doi:10.1002/1098-2426(200103)17:2<137::AID-NUM4>3.0.CO;2-0 W. F. Mitchell. A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220:350–364, 2013. doi:10.1016/j.amc.2013.05.068
@article{10356,
title = {A stabilized mixed finite element method for Poisson problem based on a three-field formulation},
journal = {ANZIAM Journal},
volume = {56},
year = {2016},
doi = {10.21914/anziamj.v57i0.10356},
language = {EN},
url = {http://dml.mathdoc.fr/item/10356}
}
Ilyas, Muhammad; Lamichhane, Bishnu Prasad. A stabilized mixed finite element method for Poisson problem based on a three-field formulation. ANZIAM Journal, Tome 56 (2016) . doi : 10.21914/anziamj.v57i0.10356. http://gdmltest.u-ga.fr/item/10356/