  We present a new mixed finite element method for a class of parabolic
equations with $p$-Laplacian and nonlinear memory. The applicability, stability
and convergence of the method are studied. First, the problem is written in a
mixed formulation as a system of one parabolic equation and a Volterra
equation. Then, the system is discretized in the space variable using the
finite element method with Lagrangian basis of degree $r\geq1$. Finally, the
Cranck-Nicolson method with the trapezoidal quadrature is applied to discretize
the time variable. For each method, we establish existence, uniqueness and
regularity of the solutions. The convergence order is found to be dependent on
the parameter $p$ on the $p$-Laplacian in the sense that it decreases as $p$
increases.
