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October  2020, 14(5): 797-818. doi: 10.3934/ipi.2020037

Numerical recovery of magnetic diffusivity in a three dimensional spherical dynamo equation

Djemaa Messaoudi 1, , Osama Said Ahmed 2, , Komivi Souley Agbodjan 3, , Ting Cheng 4, and  Daijun Jiang 4,,

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

* Corresponding author: Daijun Jiang

Received  November 2019 Revised  April 2020 Published  July 2020

Fund Project: The fourth author is supported by National Natural Science Foundation of China (Nos. 11871240 and 11771170) and Fundamental Research Funds for the Central Universities CCNU19TD010. The fifth author is supported by National Natural Science Foundation of China (Nos. 11871240 and 11401241), NSFC-RGC (China-Hong Kong, No. 11661161017) and self-determined research funds of CCNU from the colleges' basic research and operation of MOE (No. CCNU20TS003)

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.

Keywords: Inverse problems, parameter recovery, spherical dynamo model, finite element method, Tikhonov regularization.
Mathematics Subject Classification: Primary: 35R35, 49J40; Secondary: 65M32.
Citation: Djemaa Messaoudi, Osama Said Ahmed, Komivi Souley Agbodjan, Ting Cheng, Daijun Jiang. Numerical recovery of magnetic diffusivity in a three dimensional spherical dynamo equation. Inverse Problems and Imaging, 2020, 14 (5) : 797-818. doi: 10.3934/ipi.2020037
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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