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

A novel technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a quasilinear parabolic operator-differential equation with initial data $u_0\in H$. The technique is based on the smoothing property of solutions to the differential problem for $t>0$. Under certain conditions on the nonlinear term, a new estimate of order $O(\sqrt\tau+h)$ for the convergence rate in a weighted energy norm is obtained without using a priori assumptions on the additional smoothness of weak solutions.