Estimation of the Errors of the Bubnov–Galerkin Method

In Sect. 3.1, an abstract coupled problem is considered and a few theorems related to the estimation of the accuracy of the Bubnov–Galerkin method are formulated and proved. The error estimates hold for a system of differential equations of a rather general form with homogeneous boundary conditions, which corresponds to coupled thermoelastic problems for plates and shallow shells with variable thickness. In addition, a particular case of this problem (with nonhomogeneous initial conditions), where a prior estimate of the errors of the Bubnov–Galerkin method is most effective, is illustrated and discussed. Finally, a prior estimate for the Bubnov–Galerkin method to a problem generalizing a class of dynamical problems of elasticity (without a heat transfer equation) for both three-dimensional and thin-walled elements of structures is given.