Geodesic
Three-dimensional numerical simulation of the inverse problem of thermal convection using the quasi-reversibility method
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 12, pp. 2277-2288

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Inverse (time-reverse) simulation of three-dimensional thermoconvective flows is considered for a highly viscous incompressible fluid with temperature-dependent density and viscosity. The model of the fluid dynamics is described by the Stokes equations, the incompressibility and heat balance equations subject to the appropriate initial and boundary conditions. To solve the problem backward in time, the quasi-reversibility method is applied to the heat balance equation. The numerical solution is based on the introduction of a two-component vector potential for the velocity of the medium, on the application of the finite element method with a special tricubic spline basis for computing this potential, and on the application of the splitting method and the method of characteristics for computing the temperature. The numerical algorithm is designed to be executed on parallel computers. The proposed numerical algorithm is used to reconstruct the evolution of diapiric structures in the Earth's upper mantle. The computational efficiency of the algorithm is analyzed on the basis of the appropriate functionals of residuals. 
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     author = {A. T. Ismail-zade and A. I. Korotkii and I. A. Tsepelev},
     title = {Three-dimensional numerical simulation of the inverse problem of thermal convection using the quasi-reversibility method},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2277--2288},
     publisher = {mathdoc},
     volume = {46},
     number = {12},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_12_a14/}
}
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A. T. Ismail-zade; A. I. Korotkii; I. A. Tsepelev. Three-dimensional numerical simulation of the inverse problem of thermal convection using the quasi-reversibility method. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 12, pp. 2277-2288. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_12_a14/