scholarly journals Mixed Finite Element Methods for the Poisson Equation Using Biorthogonal and Quasi-Biorthogonal Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Bishnu P. Lamichhane
Keyword(s):  
Finite Element ◽  
Poisson Equation ◽  
Lagrange Multiplier ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Biorthogonal Systems ◽  
A Priori Error Estimate ◽  
The Poisson Equation ◽  
Mixed Formulations ◽  
Multiplier Space

We introduce two three-field mixed formulations for the Poisson equation and propose finite element methods for their approximation. Both mixed formulations are obtained by introducing a weak equation for the gradient of the solution by means of a Lagrange multiplier space. Two efficient numerical schemes are proposed based on using a pair of bases for the gradient of the solution and the Lagrange multiplier space forming biorthogonal and quasi-biorthogonal systems, respectively. We also establish an optimal a priori error estimate for both finite element approximations.

Higher Order Mixed FEM for the Obstacle Problem of the p-Laplace Equation Using Biorthogonal Systems

2019 ◽  
Vol 19 (2) ◽  
pp. 169-188 ◽  
Author(s):  
Lothar Banz ◽  
Bishnu P. Lamichhane ◽  
Ernst P. Stephan
Keyword(s):  
Finite Element ◽  
Error Estimate ◽  
Lagrange Multiplier ◽  
Obstacle Problem ◽  
Mixed Finite Element ◽  
Biorthogonal System ◽  
Numerical Results ◽  
Posteriori Error Estimate ◽  
A Priori Error Estimate ◽  
Polynomial Degree

AbstractWe consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for {p\in(1,\infty)}, where the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that the variational inequality can be realized in the point-wise form. We provide a general a posteriori error estimate for adaptivity and prove an a priori error estimate. We present numerical results for the adaptive scheme (mesh-size adaptivity with and without polynomial degree adaptation) for the singular case {p=1.5} and the degenerated case {p=3}. We also present numerical results on the mesh independency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution.

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

2020 ◽  
Vol 61 ◽  
pp. C197-C213
Author(s):  
Muhammad Ilyas ◽  
Bishnu P. Lamichhane
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
The Finite Element Method ◽  
Biorthogonal Systems ◽  
Poisson Problem ◽  
Engineering Mathematics ◽  
Element Method

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. References I. 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&lang=en&. D. Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge, UK, 3rd edition edition, 2007. doi:10.1017/CBO9780511618635. J. 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. J. 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. M. Ilyas. Finite element methods and multi-field applications. PhD thesis, University of Newcastle, 2019. http://hdl.handle.net/1959.13/1403421. M. 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. M. 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. 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 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. 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. Meth. Appl. Mech. Eng. 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008. J. 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. A. 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.

Sign in / Sign up

Export Citation Format

Share Document