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 on a form suitable for use in numerical analysis and with explicit dependence on the Péclet number. A stabilized finite element m...
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| Abstract | The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on 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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| AbstractList | The numerical approximation of an inverse problem subject to the
convection--diffusion equation when diffusion dominates is studied. We derive
Carleman estimates that are on 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. The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on 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. |
| Author | Burman, Erik Oksanen, Lauri Nechita, Mihai |
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| BackLink | https://doi.org/10.48550/arXiv.1811.00431$$DView paper in arXiv https://doi.org/10.1007/s00211-019-01087-x$$DView published paper (Access to full text may be restricted) |
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| Snippet | The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman... The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman... |
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| SubjectTerms | Approximation Computer Science - Numerical Analysis Convection-diffusion equation Dependence Diffusion Divergence Estimates Finite element analysis Finite element method Inverse problems Mathematical analysis Mathematics - Analysis of PDEs Mathematics - Numerical Analysis Nonlinear programming Norms Numerical analysis Numerical stability Upper bounds |
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| Title | A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime |
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