Critical Study on Finite Element Solution of a Boundary Value Problems for Equation Gravity Gyroscopic Waves in the Time Domain

In the present work, on the base of a numerical finite element method, the solution of the Dirichlet and
Neumann problems with respect to the oscillation equation for gravity-gyroscopic waves is discussed. The
approximation with respect to spatial variables is achieved by using linear splines, and the approximation with
respect to time is achieved by using cubic Hermitean splines. It is demonstrated that the use of such
approximation with respect to time allows the quality of the solution to be essentially improved as compared
with the traditional approximation ensuring the second order accuracy. The stability and accuracy of the method
are estimated. Using the method of regularization with spectrum shift, a new method is developed for solving
the spatial operator degeneration problem associated with the Neumann problem. The results of the numerical
calculations performed provide the possibility to make conclusions on the mode of behavior of the solution of
the Neumann problem depending on the problem variables.