A stabilized finite element method for convection-diffusion problems

2011 ◽  
Vol 28 (6) ◽  
pp. 1916-1943 ◽  
Author(s):  
A. Serghini Mounim
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stabilized Finite Element ◽  
Convection Diffusion ◽  
Stabilized Finite Element Method ◽  
Diffusion Problems ◽  
Element Method

A parameter-free stabilized finite element method for scalar advection-diffusion problems

2013 ◽  
Vol 11 (8) ◽  
Author(s):  
Pavel Bochev ◽  
Kara Peterson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
One Dimensional ◽  
Stabilized Finite Element ◽  
Edge Elements ◽  
Stabilized Finite Element Method ◽  
Stabilized Methods ◽  
Advection Diffusion ◽  
Diffusion Problems ◽  
Element Method

AbstractWe formulate and study numerically a new, parameter-free stabilized finite element method for advection-diffusion problems. Using properties of compatible finite element spaces we establish connection between nodal diffusive fluxes and one-dimensional diffusion equations on the edges of the mesh. To define the stabilized method we extend this relationship to the advection-diffusion case by solving simplified one-dimensional versions of the governing equations on the edges. Then we use H(curl)-conforming edge elements to expand the resulting edge fluxes into an exponentially fitted flux field inside each element. Substitution of the nodal flux by this new flux completes the formulation of the method. Utilization of edge elements to define the numerical flux and the lack of stabilization parameters differentiate our approach from other stabilized methods. Numerical studies with representative advection-diffusion test problems confirm the excellent stability and robustness of the new method. In particular, the results show minimal overshoots and undershoots for both internal and boundary layers on uniform and non-uniform grids.

scholarly journals NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS BY THE STABILIZED FINITE ELEMENT METHOD

2013 ◽  
Vol 12 (1) ◽  
pp. 67
Author(s):  
V. D. Pereira ◽  
E. C. Romão ◽  
J. B. C. Silva ◽  
L. F. M. De Moura
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Incompressible Flows ◽  
Analytical Techniques ◽  
Numerical Approach ◽  
Galerkin Finite Element ◽  
Stabilized Finite Element ◽  
Convection Diffusion ◽  
Stabilized Finite Element Method ◽  
Element Method

The fast progress has been observed in the development of numerical and analytical techniques for solving convection-diffusion and fluid mechanics problems. Here, a numerical approach, based in Galerkin Finite Element Method with Finite Difference Method is presented for the solution of a class of non-linear transient convection-diffusion problems. Using the analytical solutions and the L2 and L∞ error norms, some applications is carried and valuated with the literature.

scholarly journals A stabilized finite element method for inverse problems subject to the convection–diffusion equation. I: diffusion-dominated regime

2019 ◽  
Vol 144 (3) ◽  
pp. 451-477 ◽  
Author(s):  
Erik Burman ◽  
Mihai Nechita ◽  
Lauri Oksanen
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Approximation Order ◽  
Stability Estimates ◽  
Stabilized Finite Element ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Stabilized Finite Element Method ◽  
Element Method

AbstractThe numerical approximation of an inverse problem subject to the convection–diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are of a form suitable for use in numerical analysis and with explicit dependence on the Péclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $$H^1$$H1- or $$L^2$$L2-norms that are optimal with respect to the approximation order, the problem’s stability and perturbations in data. The convergence order is the same for both norms, but the $$H^1$$H1-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.

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