Abstract
A quasilinear parabolic problem with a time fractional derivative of the Caputo type and mixed boundary conditions is considered. The coefficients of the elliptic operator depend on the gradient of the solution, and this operator is uniformly monotone and Lipschitz-continuous. For this problem, unconditionally stable linear regularized semi-discrete scheme is constructed based on the L1-approximation of the fractional time derivative. Stability estimates are obtained by the variational method. Accuracy estimates are given provided that the initial data and the solution to the differential problem are sufficiently smooth. The proved result of stability of the semi-discrete scheme is valid for the mesh scheme obtained from the semi-discrete problem using the finite element method in spatial variables.
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