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 BackwardEuler 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.
Highlights
We will call degenerate parabolic equation an elliptic-parabolic equation of the form ( [20, Chapter III], [22, Section 44], [16]):d (Ru(t)) + A(t)u(t) = f (t), dt (1.1)Copyright c 2022 The Author(s)
The main goal of this article is precisely to provide a general theory for the mathematical analysis of a fully-discrete finite element approximation for an abstract degenerate parabolic equation
We introduce the standard terms to obtain the error estimates for parabolic problems
Summary
We will call degenerate parabolic equation an elliptic-parabolic equation of the form ( [20, Chapter III], [22, Section 44], [16]):. The mathematical analysis for the numerical approximations by finite element methods, including existence and uniqueness of the discrete solutions and quasi-optimal error estimates, has been only performed for particular degenerate parabolic equations. The main goal of this article is precisely to provide a general theory for the mathematical analysis of a fully-discrete finite element approximation for an abstract degenerate parabolic equation. We show some numerical results that confirm the expected convergence of the method according to the theory
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