• Previous Article
    Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation
  • IPI Home
  • This Issue
  • Next Article
    Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies
October  2020, 14(5): 797-818. doi: 10.3934/ipi.2020037

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

* 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.

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. BaoY. Z. CaoY. 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. ChanK. 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. HaoF. D. KangJ. 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. SchrinnerK.-H. RädlerD. SchmittM. 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. BaoY. Z. CaoY. 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. ChanK. 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. HaoF. D. KangJ. 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. SchrinnerK.-H. RädlerD. SchmittM. 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.

[1]

Na Peng, Jiayu Han, Jing An. An efficient finite element method and error analysis for fourth order problems in a spherical domain. Discrete and Continuous Dynamical Systems - B, 2022, 27 (11) : 6807-6821. doi: 10.3934/dcdsb.2022021

[2]

Abhishake Rastogi. Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4111-4126. doi: 10.3934/cpaa.2020183

[3]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

[4]

Zhong-Ci Shi, Xuejun Xu, Zhimin Zhang. The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 163-182. doi: 10.3934/dcdsb.2008.9.163

[5]

Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems and Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271

[6]

Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems and Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1

[7]

Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems and Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947

[8]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[9]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure and Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[10]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[11]

Jie Chen, Maarten de Hoop. The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter. Inverse Problems and Imaging, 2016, 10 (3) : 641-658. doi: 10.3934/ipi.2016015

[12]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024

[13]

Fengmin Xu, Yanfei Wang. Recovery of seismic wavefields by an lq-norm constrained regularization method. Inverse Problems and Imaging, 2018, 12 (5) : 1157-1172. doi: 10.3934/ipi.2018048

[14]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial and Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421

[15]

Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems and Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008

[16]

Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711

[17]

Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 91-118. doi: 10.3934/dcdss.2015.8.91

[18]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control and Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183

[19]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[20]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, 2021, 29 (3) : 2517-2532. doi: 10.3934/era.2020127

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (360)
  • HTML views (190)
  • Cited by (0)

[Back to Top]