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Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies
Numerical recovery of magnetic diffusivity in a three dimensional spherical dynamo equation
1. | School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, Kasdi Merbah University Ouargla-Algeria |
2. | School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, Department of mathematics, University of EL-Imam El-mahdi.Kosti-Sudan |
3. | School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, University of Lome, TOGO |
4. | School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China |
This paper is concerned with the analysis on a numerical recovery of the magnetic diffusivity in a three dimensional (3D) spherical dynamo equation. We shall transform the ill-posed problem into an output least squares nonlinear minimization by an appropriately selected Tikhonov regularization, whose regularizing effects and mathematical properties are justified. The nonlinear optimization problem is approximated by a fully discrete finite element method and its convergence shall be rigorously established.
References:
[1] |
G. Bao, Y. Z. Cao, Y. L. Hao and and K. Zhang,
First order second moment analysis for the stochastic interface grating problem, J Sci. Comput., 77 (2018), 419-442.
doi: 10.1007/s10915-018-0712-z. |
[2] |
P. Cardin and P. Olson,
An experimental approach to thermochemical convection in the Earth's core, Geophys. Res. Lett., 19 (1992), 1995-1998.
doi: 10.1029/92GL01883. |
[3] |
K. H. Chan, K. Zhang and J. Zou,
Spherical interface dynamos: Mathematical theory, finite element approximation, and application, SIAM J. Numer. Anal., 44 (2006), 1877-1902.
doi: 10.1137/050635596. |
[4] |
Z.M. Chen and J. Zou,
An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems, SIAM J. Control Optim., 37 (1999), 892-910.
doi: 10.1137/S0363012997318602. |
[5] |
M. Fischer, G. Gerbeth, A. Giesecke and F. Stefani, Inferring basic parameters of the geodynamo from sequences of polarity reversals, Inverse Problems 25 (2009), 065011.
doi: 10.1088/0266-5611/25/6/065011. |
[6] |
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag Berlin Heidelberg, 1986.
doi: 10.1007/978-3-642-61623-5. |
[7] |
G. A. Glatzmaier and P. H. Roberts,
A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle, Phys. Earth Planet. Inter., 91 (1995), 63-75.
doi: 10.1016/0031-9201(95)03049-3. |
[8] |
Y. L. Hao, F. D. Kang, J. Z. Li and K. Zhang,
A numerical method for Maxwell's equations with random interfaces via shape calculus and pivoted low-rank approximation, J. Comput. Phys., 371 (2018), 1-18.
doi: 10.1016/j.jcp.2018.05.004. |
[9] |
Y. L. Keung and J. Zou,
Numerical identifications of parameters in parabolic systems, Inverse Problems, 14 (1998), 83-100.
doi: 10.1088/0266-5611/14/1/009. |
[10] |
W. Kuang and J. Bloxham,
Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: Weak and strong field dynamo action, J. Comput. Phys., 153 (1999), 51-81.
doi: 10.1006/jcph.1999.6274. |
[11] |
D. Liu, W. Kuang and A. Tangborn, High-order compact implicit difference methods for parabolic equations in geodynamo simulation, Adv. Math. Phys., (2009), Art. ID 568296, 23 pp.
doi: 10.1155/2009/568296. |
[12] |
M. Schrinner, K.-H. Rädler, D. Schmitt, M. Rheinhardt and U. R. Christensen,
Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo, Geophys. Astrophys. Fluid Dyn., 101 (2007), 81-116.
doi: 10.1080/03091920701345707. |
[13] |
L. R. Scott and S. Zhang,
Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), 483-493.
doi: 10.1090/S0025-5718-1990-1011446-7. |
[14] |
R. Temam,
Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II, Arch. Rat. Mech. Anal., 33 (1969), 377-385.
doi: 10.1007/BF00247696. |
[15] |
J. L. Xie and J. Zou,
Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43 (2005), 1504-1535.
doi: 10.1137/030602551. |
[16] |
J. Xu, Theory of Multilevel Methods, Ph.D thesis, Cornell University, 1989. |
[17] |
M. Yamamoto and J. Zou,
Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), 1181-1202.
doi: 10.1088/0266-5611/17/4/340. |
[18] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, 2, Springer-Verlag, New York, (1985).
doi: 10.1007/978-1-4612-5020-3. |
show all references
References:
[1] |
G. Bao, Y. Z. Cao, Y. L. Hao and and K. Zhang,
First order second moment analysis for the stochastic interface grating problem, J Sci. Comput., 77 (2018), 419-442.
doi: 10.1007/s10915-018-0712-z. |
[2] |
P. Cardin and P. Olson,
An experimental approach to thermochemical convection in the Earth's core, Geophys. Res. Lett., 19 (1992), 1995-1998.
doi: 10.1029/92GL01883. |
[3] |
K. H. Chan, K. Zhang and J. Zou,
Spherical interface dynamos: Mathematical theory, finite element approximation, and application, SIAM J. Numer. Anal., 44 (2006), 1877-1902.
doi: 10.1137/050635596. |
[4] |
Z.M. Chen and J. Zou,
An augmented Lagrangian method for identifying discontinuous parameters in elliptic systems, SIAM J. Control Optim., 37 (1999), 892-910.
doi: 10.1137/S0363012997318602. |
[5] |
M. Fischer, G. Gerbeth, A. Giesecke and F. Stefani, Inferring basic parameters of the geodynamo from sequences of polarity reversals, Inverse Problems 25 (2009), 065011.
doi: 10.1088/0266-5611/25/6/065011. |
[6] |
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag Berlin Heidelberg, 1986.
doi: 10.1007/978-3-642-61623-5. |
[7] |
G. A. Glatzmaier and P. H. Roberts,
A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle, Phys. Earth Planet. Inter., 91 (1995), 63-75.
doi: 10.1016/0031-9201(95)03049-3. |
[8] |
Y. L. Hao, F. D. Kang, J. Z. Li and K. Zhang,
A numerical method for Maxwell's equations with random interfaces via shape calculus and pivoted low-rank approximation, J. Comput. Phys., 371 (2018), 1-18.
doi: 10.1016/j.jcp.2018.05.004. |
[9] |
Y. L. Keung and J. Zou,
Numerical identifications of parameters in parabolic systems, Inverse Problems, 14 (1998), 83-100.
doi: 10.1088/0266-5611/14/1/009. |
[10] |
W. Kuang and J. Bloxham,
Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: Weak and strong field dynamo action, J. Comput. Phys., 153 (1999), 51-81.
doi: 10.1006/jcph.1999.6274. |
[11] |
D. Liu, W. Kuang and A. Tangborn, High-order compact implicit difference methods for parabolic equations in geodynamo simulation, Adv. Math. Phys., (2009), Art. ID 568296, 23 pp.
doi: 10.1155/2009/568296. |
[12] |
M. Schrinner, K.-H. Rädler, D. Schmitt, M. Rheinhardt and U. R. Christensen,
Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo, Geophys. Astrophys. Fluid Dyn., 101 (2007), 81-116.
doi: 10.1080/03091920701345707. |
[13] |
L. R. Scott and S. Zhang,
Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), 483-493.
doi: 10.1090/S0025-5718-1990-1011446-7. |
[14] |
R. Temam,
Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II, Arch. Rat. Mech. Anal., 33 (1969), 377-385.
doi: 10.1007/BF00247696. |
[15] |
J. L. Xie and J. Zou,
Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43 (2005), 1504-1535.
doi: 10.1137/030602551. |
[16] |
J. Xu, Theory of Multilevel Methods, Ph.D thesis, Cornell University, 1989. |
[17] |
M. Yamamoto and J. Zou,
Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), 1181-1202.
doi: 10.1088/0266-5611/17/4/340. |
[18] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, 2, Springer-Verlag, New York, (1985).
doi: 10.1007/978-1-4612-5020-3. |
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