Correctness of an operator-differential scheme and substantiation of the Galerkin method for hyperbolic equations

A theorem on the conditional correctness of an operator-differential scheme is proved. Using this theorem, the Galerkin method for an abstract quasilinear hyperbolic equation is substantiated in the case when the coercive solvability conditions are absent and the existence of the sufficiently smooth exact solution is supposed. The unique solvability of the approximate problems is stated and the error estimate exact by the order of approximation is obtained. The use of these results is illustrated by an example of finite element schemes applied to the first initial boundary value problem for a second-order hyperbolic equation.