Projection-difference method with the Crank-Nicolson scheme in time for the approximate solution of a parabolic equation with an integral condition for the solution

Abstract

In a separable Hilbert space, we consider an abstract parabolic problem with a nonlocal integral condition on the solution. This problem is solved approximately by a projection-difference method. The spatial discretization of the problem is performed by the Galerkin method, and for the time discretization, we use the Crank-Nicolson scheme. In the case of a certain smooth solvability of the exact problem, we obtain efficient estimates for the accuracy of the approximate solutions. Those estimates imply the convergence of the error to zero up to the second order in time. In addition, these estimates accurately take into account the approximation properties of the projection subspaces; this is observed on subspaces of the type of finite elements in the derivation of the convergence rate of the method.