3,291 research outputs found
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-. Fredi Troltzsch. Optimal control of partial differential equations, volume 112. American Mathematical Society, 2010. http://www.ams.org/books/gsm/112/
Primal dual mixed finite element methods for indefinite advection--diffusion equations
We consider primal-dual mixed finite element methods for the
advection--diffusion equation. For the primal variable we use standard
continuous finite element space and for the flux we use the Raviart-Thomas
space. We prove optimal a priori error estimates in the energy- and the
-norms for the primal variable in the low Peclet regime. In the high
Peclet regime we also prove optimal error estimates for the primal variable in
the norm for smooth solutions. Numerically we observe that the method
eliminates the spurious oscillations close to interior layers that pollute the
solution of the standard Galerkin method when the local Peclet number is high.
This method, however, does produce spurious solutions when outflow boundary
layer presents. In the last section we propose two simple strategies to remove
such numerical artefacts caused by the outflow boundary layer and validate them
numerically.Comment: 25 pages, 6 figures, 5 table
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
The nonconforming virtual element method for eigenvalue problems
We analyse the nonconforming Virtual Element Method (VEM) for the
approximation of elliptic eigenvalue problems. The nonconforming VEM allow to
treat in the same formulation the two- and three-dimensional case.We present
two possible formulations of the discrete problem, derived respectively by the
nonstabilized and stabilized approximation of the L^2-inner product, and we
study the convergence properties of the corresponding discrete eigenvalue
problem. The proposed schemes provide a correct approximation of the spectrum,
in particular we prove optimal-order error estimates for the eigenfunctions and
the usual double order of convergence of the eigenvalues. Finally we show a
large set of numerical tests supporting the theoretical results, including a
comparison with the conforming Virtual Element choice
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