Error estimates of the Galerkin method for a weakly solvable parabolic equation with a nonlocal-in-time condition for the solution

Abstract

We consider the abstract linear parabolic equation with a nonlocal-in-time condition for an integral-type solution in a separable Hilbert space. The problem is solved approximately using the semidiscrete Galerkin method. Under conditions of weak solvability for the problem, we establish error estimates for an approximate solution. Under additional assumptions on the smoothness of the solution of the exact problem, we also obtain the convergence rate that is exact in the order of approximation for projection subspaces of finite-element type.