Weighted estimate for the convergence rate of a projection difference scheme for a quasilinear parabolic equation

Projection–difference method for controlled Fourier filtering

“…This method makes it possible to investigate the convergence of approximations of FDE and optimal control problems for natural smoothness classes of FDE solutions. A bound on the rate of convergence of the method for an abstract quasi-linear parabolic FDE in a weighted energy norm has been obtained in [9]. In the present article we apply the results of [9] to construct a projection-difference scheme (PDS) for the Fourier-filtering problem and to prove convergence in the functional of projection-difference approximations of the Fourier-filter control problem.…”

“…A bound on the rate of convergence of the method for an abstract quasi-linear parabolic FDE in a weighted energy norm has been obtained in [9]. In the present article we apply the results of [9] to construct a projection-difference scheme (PDS) for the Fourier-filtering problem and to prove convergence in the functional of projection-difference approximations of the Fourier-filter control problem. There have been no previous studies of the convergence of the projection-difference method for Fourierfilter control problems.…”

“…We prove convergence in the functional of the finite-dimensional approximating problems to the original Fourier-filter control problem without prior assumptions on the smoothness of the solutions; the achieved order of convergence is the same as for the PDS. The concluding section (Section 6) reviews the propositions from [9] that are used in previous sections.…”

“…Yet practical computation of Fourier filters is impossible without finite-dimensional approximation of FDE and corresponding optimal control problems. For purposes of such approximation we propose using the projection-difference method that we developed in [8,9]. This method makes it possible to investigate the convergence of approximations of FDE and optimal control problems for natural smoothness classes of FDE solutions.…”

“…Section 4 constructs the PDS for the Fourier filtering quasi-linear equation. The PDS is concretized for the abstract operator equation from [9], where all the finite-dimensional operators entering the PDS are easily written out. We prove the convergence of the iteration method for the nonlinear scheme and derive a constructive bound on the rate of convergence of the PDS in the weighted energy norm.…”