Geodesic
A mixed finite element method for a~shallow shell problem with a~stress function
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2009), pp. 13-22

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In this paper we consider the problem of a deflected mode of a shallow shell. The stress function and the normal component of the displacement of the median surface of the shell are unknown functions. We propose a mixed variational statement of the problem, where the second derivatives of the stress function and the normal component of the displacement of the median surface are additional unknowns. This enables us to construct the finite element approximation of the initial problem. We prove the existence of a unique solution of the approximating problem and estimate the rate of convergence of the discrete solution.
Keywords: shallow shell, mixed finite element method, errors estimates. 
@article{IVM_2009_5_a1,
     author = {V. V. Verbitskii and A. V. Verbitskii},
     title = {A mixed finite element method for a~shallow shell problem with a~stress function},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {13--22},
     publisher = {mathdoc},
     number = {5},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_5_a1/}
}
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V. V. Verbitskii; A. V. Verbitskii. A mixed finite element method for a~shallow shell problem with a~stress function. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2009), pp. 13-22. http://geodesic.mathdoc.fr/item/IVM_2009_5_a1/