Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

TL;DR

This paper develops an abstract finite element framework for analyzing degenerate parabolic mixed equations, providing conditions for unique solutions, error estimates, and applications to fluid dynamics and electromagnetism.

Contribution

It introduces a unified theoretical approach for fully discrete finite element approximations of degenerate parabolic mixed equations, including error analysis and practical applications.

Findings

Proved existence and uniqueness of solutions for the fully discrete scheme.

Derived quasi-optimal error estimates for the approximation.

Validated the theory with numerical experiments.

Abstract

The aim of this work is to show an abstract framework to analyze the numerical approximation for a family of linear degenerate parabolic mixed equations by using a finite element method in space and a Backward-Euler scheme in time. We consider sufficient conditions to prove that the fully-discrete problem has a unique solution and prove quasi-optimal error estimates for the approximation. Furthermore, we show that mixed finite element formulations arising from dynamics fluids (time-dependent Stokes problem) and from electromagnetic applications (eddy current models), can be analyzed as applications of the developed theory. Finally, we include numerical tests to illustrate the performance of the method and confirm the theoretical results.