A Mixed Finite Element Method for a Class of Evolution Differential Equations with $p$-Laplacian and Memory

TL;DR

This paper introduces a new mixed finite element method for solving parabolic equations with p-Laplacian and nonlinear memory, analyzing its stability, convergence, and how the parameter p affects the convergence order.

Contribution

The paper develops a novel mixed finite element approach for a class of evolution equations with p-Laplacian and memory, including stability and convergence analysis.

Findings

Convergence order decreases as p increases.

Existence, uniqueness, and regularity of solutions are established.

Method is applicable to nonlinear parabolic equations with memory.

Abstract

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.