Numerical solution of the Stokes problem under free boundary by the modified projection gradient method
DOI:
https://doi.org/10.7242/1999-6691/2008.1.1.8Keywords:
Abstract
From a variational standpoint, the solution of Stokes equation is reduced to the constrained minimization of the total energy functional over the space of solenoidal fields. The paper presents the finite element method combined with the gradient projection method to obtain the approximate solution of this problem by the unconstrained minimization of the quadratic functional with a reduced number of unknown variables, which has not been previously used for solving such a problem. The numerical solutions of a few test free boundary problems are presented to reveal the advantages of the developed method over the penalty and Lagrangian methods with respect to the accuracy, stability, and computer speed.
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Grigor’ev A.S., Sahmuralova Z.E. Teoreticeskoe opredelenie skorostej dvizenia dnevnoj poverhnosti pri nekotoryh mehanizmah deformirovania zemnoj kory // Sovremennye dvizenia zemnoj kory. - Tartu, 1973. - Vyp. 5. - S. 595-603.
Danilin U.M. O minimizacii funkcii v zadacah s ograniceniami tipa ravenstv // Kibernetika. - 1971. - No 2.- S. 88-95.
Kobel’kov G.M. O svedenii kraevoj zadaci dla bigarmoniceskogo uravnenia k zadace tipa Stoksa // Dokl. AN SSSR. - 1985.- T. 283, No 1. - S. 539-542.
Malevskij A.V. Mehanika processa obrazovania astenosfernogo diapira // Cislennoe modelirovanie i analiz geofiziceskih processov: Vycisl. sejsmologia: vyp. 20. - M.: Nauka, 1987. - S. 52-60
Marcuk G.I. Metody vycislitel’noj matematiki. - Novosibirsk: Nauka, 1978. - 536 s.
Nadai A. Plasticnost’ i razrusenie tverdyh tel. - M.: IL, 1954. - 863 s.
Pak V.V. Cislennaa model’ dvuhfaznoj sredy so slaboszimaemym skeletom i nekotorye geofiziceskie prilozenia // Dal’nevostocnyj matematiceskij zurnal. - 2007.- T. 7, No 1-2. - S. 79-90.
Pak V.V. Mnogotemperaturnaa model’ kompakcii magmaticeskogo rasplava v astenosfere (Cislennyj podhod) // Fizika Zemli. - 2007. - No 9. - S. 79-86.
Temam R. Uravnenia Nav’e-Stoksa. Teoria i cislennyj analiz. - M.: Mir, 1981.- 408 s.
Timuhin G.I. O postroenii nekotoryh shem priblizennogo re- senia uravnenij dinamiki vazkoj neszimaemoj zidkosti // Cisl. metody mehan. splos. sr. - Novosibirsk: Nauka, 1972.- T. 3, No 2. - C. 77-85.
Han, Hou-de. An analysis of penalty nonconforming finite element method for Stokes equations // J. Comput. Math. - 1986. - V. 4, No 2. - P. 164-172.
Loewenherz D.S. and Lawrence C.J., The effect of viscosity stratification on the instability of a free surface flow at low Reynolds number // Phys. of fluids A 1. - 1989. - V. 10. - P. 1686 - 1693. DOI
Lukas D, Dostal Z. Optimal multigrid preconditioned semi-monotonic augmented Lagrangians applied to the Stokes problem // Numerical linear algebra with applications. - 2007. - V. 14, No 9. - P. 741-750. DOI
Schmeling H.G., Alexander R., Marquari G. Finite deformation in and around a fluid sphere moving through a viscous medium: implications for diapic ascent. // Tectonophysics. - 1988, - V. 149, No 1-2. - P. 17-34. DOI
Wieners C. Robust multigrid methods for nearly incompressible elasticity // Computing. - 2000. - V. 64, No 4. - P. 289-306. DOI
Woidt W.D. Neugebauer H.J. Finite element models of density instability by means of bicubic spline interpolation // Phys. Earth and Planet. Inter. - 1980. - V.21, No 2/3. - P. 176-180. DOI
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