Estimates of error of semidiscrete approximations by Galerkin for parabolic equations with boundary condition of Neumann type (Q677790)

The problem \[ u'(t)+A(t)u(t)+ B(t)u(t)= f(t),\qquad u(0)=u^0 \] is considered in a separable Hilbert space \(H\) on the time segment \(\langle 0,T\rangle\), where \(A(t)\) is unbounded selfadjoint and positive definite for every \(t\in\langle 0,T\rangle\), in case of the domain of definition of the operator \(A^{1/2}(t)\) does not depend on \(t\). The approximation of the given problem in finite-dimensional subspaces, called semidiscrete approximation by Galerkin, is described. The paper is devoted to an estimation of the errors of the approximate solutions in subspaces of finite-element type under the condition of generalized solvability of the exact problem. An error estimation under the condition of smooth solvability of the exact problem is also given.