Fully-discrete finite element approximation for a family of degenerate parabolic problems

TL;DR

This paper develops an abstract framework for analyzing finite element and Backward-Euler approximations of degenerate parabolic problems, ensuring well-posedness, uniqueness, and error estimates, with applications to electromagnetic models.

Contribution

It introduces a novel abstract approach for fully-discrete finite element approximation of degenerate parabolic equations, including error analysis and application to electromagnetic problems.

Findings

Proved existence and uniqueness of solutions under certain conditions.

Derived quasi-optimal error estimates for the numerical approximation.

Validated the theory with numerical tests on electromagnetic applications.

Abstract

The aim of this work is to show an abstract framework to analyze the numerical approximation by using a finite element method in space and a Backward-Euler scheme in time of a family of degenerate parabolic problems. We deduce sufficient conditions to ensure that the fully-discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally, we show a degenerate parabolic problem which arises from electromagnetic applications and deduce its well-posedness and convergence by using the developed abstract theory, including numerical tests to illustrate the performance of the method and confirm the theoretical results. Keywords: parabolic degenerate equations, parabolic-elliptic equations, finite element method, backward Euler scheme, fully-discrete approximation, error estimates, eddy current model.