On the finite element method for a nonlocal degenerate parabolic problem

TL;DR

This paper develops and analyzes a finite element method for solving nonlinear nonlocal degenerate parabolic equations, providing convergence proofs, error bounds, and numerical tests with explicit solutions.

Contribution

It introduces a linearized Crank-Nicolson-Galerkin finite element approach with polynomial degree k≥1 for this class of equations, including convergence and error analysis.

Findings

Convergence and error bounds are established for the proposed method.

Explicit solutions are used to validate the numerical implementation.

The method performs well in Matlab tests with polynomial approximations.

Abstract

The aim of this paper is the numerical study of a class of nonlinear nonlocal degenerate parabolic equations. The convergence and error bounds of the solutions are proved for a linearized Crank-Nicolson-Galerkin finite element method with polynomial approximations of degree $k\geq 1$. Some explicit solutions are obtained and used to test the implementation of the method in Matlab environment.