# American Institute of Mathematical Sciences

2015, 2015(special): 525-532. doi: 10.3934/proc.2015.0525

## A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems

 1 Departamento de Matemáticas, Universidade da Coruña, Campus de Elviña s/n 15071 A Coruña, Spain 2 Basque Center for Applied Mathematics, Alameda Mazarredo 14, 48009 Bilbao, Spain 3 Department of Computing, Mathematics and Physics, Bergen University College, Bergen, Norway

Received  September 2014 Revised  September 2015 Published  November 2015

We present an augmented dual-mixed variational formulation for a linear convection-diffusion equation with homogeneous Dirichlet boundary conditions. The approach is based on the addition of suitable least squares type terms. We prove that for appropriate values of the stabilization parameters, that depend on the diffusion coefficient and the magnitude of the convective velocity, the new variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds. In particular, we derive the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart-Thomas and continuous piecewise polynomials. In addition, we introduce a simple a posteriori error estimator which is reliable and locally efficient. Finally, we provide numerical experiments that illustrate the behavior of the method.
Citation: M. González, J. Jansson, S. Korotov. A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems. Conference Publications, 2015, 2015 (special) : 525-532. doi: 10.3934/proc.2015.0525
##### References:
 [1] T. P. Barrios, J. M. Cascón and M. González, A posteriori error analysis of an augmented mixed finite element method for Darcy flow, Comput. Methods Appl. Mech. Engrg., 283 (2015), 909-922. [2] J. Douglas Jr. and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52. [3] A. Ern and A. F. Stephansen, A posterior energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods, J. Comput. Math., 26 (2008), 488-510. [4] A. Masud and T. J. R. Hughes, A stabilized mixed finite element method for Darcy flow, Comput. Methods Appl. Mech. Engrg., 191 (2002), 4341-4370. [5] J. E. Roberts and J. M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions, vol. II, Finite Element Methods (Part 1). North-Holland, Amsterdam, 1991.

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##### References:
 [1] T. P. Barrios, J. M. Cascón and M. González, A posteriori error analysis of an augmented mixed finite element method for Darcy flow, Comput. Methods Appl. Mech. Engrg., 283 (2015), 909-922. [2] J. Douglas Jr. and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44 (1985), 39-52. [3] A. Ern and A. F. Stephansen, A posterior energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods, J. Comput. Math., 26 (2008), 488-510. [4] A. Masud and T. J. R. Hughes, A stabilized mixed finite element method for Darcy flow, Comput. Methods Appl. Mech. Engrg., 191 (2002), 4341-4370. [5] J. E. Roberts and J. M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions, vol. II, Finite Element Methods (Part 1). North-Holland, Amsterdam, 1991.
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