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

The 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 me...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 144; no. 3; pp. 451 - 477
Main Authors: Burman, Erik, Nechita, Mihai, Oksanen, Lauri
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2020
Springer Nature B.V
Subjects:
Approximation
Convection-diffusion equation
Diffusion
Divergence
Estimates
Finite element analysis
Finite element method
Inverse problems
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Norms
Numerical Analysis
Numerical and Computational Physics
Numerical stability
Simulation
Stability
Theoretical
Upper bounds
65N12
35J15
65N21
65N30
65N20
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Summary:The 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 - or L 2 -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 -estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-019-01087-x