scholarly journals Analysis of a Stabilized Finite Element Approximation of the Transient Convection‐Diffusion Equation Using an ALE Framework

2006 ◽  
Vol 44 (5) ◽  
pp. 2159-2197 ◽  
Author(s):  
Santiago Badia ◽  
Ramon Codina
Keyword(s):  
Finite Element ◽  
Diffusion Equation ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Stabilized Finite Element ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

scholarly journals Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

2021 ◽  
Author(s):  
Johnny Guzmán ◽  
Erik Burman
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Convection Term ◽  
Differentiation Formula ◽  
Numerical Examples ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Piecewise Affine ◽  
Time Discretisation

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection--diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank-Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the $\tau^2 + h^{p+{\frac12}}$ error estimates for the $L^2$-norm under either the standard hyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used, or in the case of finite element approximation of order $p \ge 1$, a stronger, so-called $4/3$-CFL, i.e. $\tau \leq C h^{4/3}$. The theory is illustrated with some numerical examples.

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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