A mixed finite element method for a class of evolution differential equations with p-Laplacian and memory

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 >= 1. 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. (C) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

We present a new mixed finite element method for solving a class of parabolic equations with p-Laplacian and nonlinear memory. The applicability, stability, and convergence of the method are studied. The paper provides the numerical discretization process and the relation between the convergence order and the parameter p.