Ill-Posed Problems and Stabilized Finite Element Methods
Event details
| Date | 05.11.2019 |
| Hour | 16:15 › 17:15 |
| Speaker | Prof. Erik BURMAN, University College London |
| Location |
MA A3 30
|
| Category | Conferences - Seminars |
Computational Mathematics Seminar
Abstract:
In this talk we will consider some ill-posed elliptic equations and their discretiza-
tion using nite element methods. The standard approach to ill-posed problems is to regularize
the continuous problem so that existence and uniqueness is guaranteed. The regularized prob-
lem can then be solved using standard nite element methods. When using this strategy, in
order to optimize accuracy, the regularization parameter must be chosen as a function both
of the stability properties of the ill-posed problem, the mesh parameter and perturbations in
data. Here we will propose a di erent approach [1], where the ill-posed pde is discretized in
an optimization framework, prior to regularization. To ensure discrete well-posedness we add
stabilizing terms to the formulation, drawing on experience from stabilized FEM and discon-
tinuous Galerkin methods. The error in the resulting nite element reconstructions is then
analyzed using Carleman estimates on the continuous problem. This results in approximations
that are optimal with respect to the approximation order of the nite element space and the
stability of the computed quantity. The mesh parameter here plays the role of the regular-
ization parameter. Mesh resolution can be chosen independently of the stability properties of
the physical problem, but must match perturbations in data, in a way made explicit in the
estimates. Some examples of problems analyzed in this framework will be presented, selected
from recent work on the Helmholtz equation [4], the convection{di usion equation [5], Stokes'
equations [2] and Darcy's equation [3].
References
[1] E. Burman. Stabilised nite element methods for ill-posed problems with conditional stability. Building
bridges: connections and challenges in modern approaches to numerical partial di erential equations, 93127,
Lect. Notes Comput. Sci. Eng., 114, Springer, [Cham], 2016., Dec. 2015.
[2] E. Burman, P. Hansbo, Stabilized nonconforming nite element methods for data assimilation in incom-
pressible
ows. Math. Comp. 87, no. 311, 2018.
[3] E. Burman, M. G. Larson, L. Oksanen. Primal dual mixed nite element methods for the elliptic Cauchy
problem, arXiv:1712.10172, Siam J. Num. Anal., to appear, 2018.
[4] E. Burman, M. Nechita, L. Oksanen. Unique continuation for the Helmholtz equation us-
ing stabilized nite element methods. Journal de Mathematiques Pures et Appliquees,
https://doi.org/10.1016/j.matpur.2018.10.003.
[5] E. Burman, M. Nechita, L. Oksanen. A stabilized nite element method for inverse problems subject to the
convection-di usion equation. I: di usion-dominated regime. arXiv:1811.00431, 2018.
Abstract:
In this talk we will consider some ill-posed elliptic equations and their discretiza-
tion using nite element methods. The standard approach to ill-posed problems is to regularize
the continuous problem so that existence and uniqueness is guaranteed. The regularized prob-
lem can then be solved using standard nite element methods. When using this strategy, in
order to optimize accuracy, the regularization parameter must be chosen as a function both
of the stability properties of the ill-posed problem, the mesh parameter and perturbations in
data. Here we will propose a di erent approach [1], where the ill-posed pde is discretized in
an optimization framework, prior to regularization. To ensure discrete well-posedness we add
stabilizing terms to the formulation, drawing on experience from stabilized FEM and discon-
tinuous Galerkin methods. The error in the resulting nite element reconstructions is then
analyzed using Carleman estimates on the continuous problem. This results in approximations
that are optimal with respect to the approximation order of the nite element space and the
stability of the computed quantity. The mesh parameter here plays the role of the regular-
ization parameter. Mesh resolution can be chosen independently of the stability properties of
the physical problem, but must match perturbations in data, in a way made explicit in the
estimates. Some examples of problems analyzed in this framework will be presented, selected
from recent work on the Helmholtz equation [4], the convection{di usion equation [5], Stokes'
equations [2] and Darcy's equation [3].
References
[1] E. Burman. Stabilised nite element methods for ill-posed problems with conditional stability. Building
bridges: connections and challenges in modern approaches to numerical partial di erential equations, 93127,
Lect. Notes Comput. Sci. Eng., 114, Springer, [Cham], 2016., Dec. 2015.
[2] E. Burman, P. Hansbo, Stabilized nonconforming nite element methods for data assimilation in incom-
pressible
ows. Math. Comp. 87, no. 311, 2018.
[3] E. Burman, M. G. Larson, L. Oksanen. Primal dual mixed nite element methods for the elliptic Cauchy
problem, arXiv:1712.10172, Siam J. Num. Anal., to appear, 2018.
[4] E. Burman, M. Nechita, L. Oksanen. Unique continuation for the Helmholtz equation us-
ing stabilized nite element methods. Journal de Mathematiques Pures et Appliquees,
https://doi.org/10.1016/j.matpur.2018.10.003.
[5] E. Burman, M. Nechita, L. Oksanen. A stabilized nite element method for inverse problems subject to the
convection-di usion equation. I: di usion-dominated regime. arXiv:1811.00431, 2018.
Practical information
- General public
- Free
Organizer
- Prof. Marco Picasso