A variational problem of determination of the amplitude of the velocity vector of elastic displacements and the electric potential of a piezoelectric with instantaneous memory that executes steady-state vibrations under the influence of harmonic loads of given circular frequency is formulated. We establish conditions of well-posedness of this class of problems and find a priori estimates of convergence of approximations of the finite-element method to their solution. A posteriori error estimators of approximations of the finite-element method, which enable us to determine the distribution of energy norms of errors by solving local problems of the residual of approximation at each finite element of triangulation, are constructed. An algorithm for calculating estimators of this type is described in detail for two-dimensional problems for both triangular and tetragonal finite elements, which may be extended to the three-dimensional case. The efficiency and reliability of the estimator of piecewise bilinear approximations of the finite-element method is illustrated by numerical solutions of a model problem.
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 52, No. 4, pp. 88–98, October–December, 2009.
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Chaban, F.V., Shynkarenko, H.A. A posteriori error estimators of finite-element approximations for problems of forced harmonic vibrations of piezoelectrics. J Math Sci 174, 229–242 (2011). https://doi.org/10.1007/s10958-011-0293-y
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DOI: https://doi.org/10.1007/s10958-011-0293-y