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

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 uniq...

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Published inMathematical modelling and analysis Vol. 27; no. 1; pp. 134 - 160
Main Authors Acevedo, Ramiro, Gómez, Christian, López-Rodríguez, Bibiana
Format Journal Article
LanguageEnglish
Published Vilnius Vilnius Gediminas Technical University 07.02.2022
Subjects
Approximation
backward euler scheme
eddy current model
Electromagnetism
error estimates
Finite element method
fully-discrete approximation
Mathematical analysis
parabolic degenerate equations
parabolic-elliptic equations
Colombia
parabolic degenerate equations
finite element method
parabolic-elliptic equations
backward Euler scheme
eddy current model
error estimates
fully-discrete approximation
Online AccessGet full text
ISSN1392-6292
1648-3510
DOI10.3846/mma.2022.12846

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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.
AbstractList 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.
The aim of this work is to show an 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 theory, including numerical tests to illustrate the performance of the method and confirm the theoretical results.
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. Keywords: parabolic degenerate equations, parabolic-elliptic equations, finite element method, backward Euler scheme, fully-discrete approximation, error estimates, eddy current model. AMS Subject Classification: 35K65; 35K90.
Audience Academic
Author Acevedo, Ramiro
Gómez, Christian
López-Rodríguez, Bibiana
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  givenname: Bibiana
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  surname: López-Rodríguez
  fullname: López-Rodríguez, Bibiana
  organization: Universidad Nacional de Colombia sede Medellín, Escuela de Matemáticas, Carrera 6 # 59a-110, Medellín (Antioquia), Colombia
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10.1201/9780429187810-22
10.1051/m2an/1982160201611
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Keywords parabolic degenerate equations
finite element method
parabolic-elliptic equations
backward Euler scheme
eddy current model
error estimates
fully-discrete approximation
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Snippet 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...
The aim of this work is to show an framework to analyze the numerical approximation by using a finite element method in space and a BackwardEuler scheme in...
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StartPage 134
SubjectTerms Approximation
backward euler scheme
eddy current model
Electromagnetism
error estimates
Finite element method
fully-discrete approximation
Mathematical analysis
parabolic degenerate equations
parabolic-elliptic equations
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Title Fully-discrete finite element approximation for a family of degenerate parabolic problems
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