Fully-discrete finite element approximation for a family of degenerate parabolic problems
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.
Keyword : parabolic degenerate equations, parabolic-elliptic equations, finite element method, backward Euler scheme, fully-discrete approximation, error estimates, eddy current model
How to Cite
Acevedo, R., Gómez, C., & López-Rodríguez, B. (2022). Fully-discrete finite element approximation for a family of degenerate parabolic problems. Mathematical Modelling and Analysis, 27(1), 134–160. https://doi.org/10.3846/mma.2022.12846
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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H. Ammari, A. Buffa and J.-C. Nédélec. A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math., 60(5):1805–1823, 2000. https://doi.org/10.1137/S0036139998348979
A. Bermúdez, R. Mu noz Sola, C. Reales, R. Rodríguez and P. Salgado. A transient eddy current problem on a moving domain. Mathematical analysis. SIAM J. Math. Anal., 45(6):3629–3650, 2013. https://doi.org/10.1137/130914425
A.Bermúdez, R.Mu noz Sola, C.Reales, R.Rodríguez and P.Salgado. A transient eddy current problem on a moving domain. Numerical analysis. Adv. Comput. Math.,42(4):757–789,2016. https://doi.org/10.1007/s10444-015-9441-0
A. Bermúdez, C. Reales, R. Rodríguez and P. Salgado. Numerical analysis of a transient eddy current axisymmetric problem involving velocity terms. Numer. Methods Partial Differential Equations, 28(3):984–1012, 2012. https://doi.org/10.1002/num.20670
C. Bernardi and V. Girault. A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal, 35(5):1893–1916, 1998. https://doi.org/10.1137/S0036142995293766
A. Bossavit. Computational electromagnetism. Electromagnetism. Academic Press, Inc., San Diego, CA, 1998. Variational formulations, Complementarity, Edge elements.
P.G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. https://doi.org/10.1137/1.9780898719208. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]
A. Ern and J-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004. https://doi.org/10.1007/978-1-4757-4355-5
Jr. Kuttler and L. Kenneth. A degenerate nonlinear Cauchy problem. Applicable Anal., 13(4):307–322, 1982. https://doi.org/10.1080/00036818208839402
Jr. Kuttler and L. Kenneth. The Galerkin method and degenerate evolution equations. J. Math. Anal. Appl., 107(2):396–413, 1985. https://doi.org/10.1016/0022-247X(85)90321-X
Jr. Kuttler and L. Kenneth. Time-dependent implicit evolution equations. Nonlinear Anal., 10(5):447–463, 1986. https://doi.org/10.1016/0362-546X(86)90050-7
R.C. MacCamy and M. Suri. A time-dependent interface problem for two-dimensional eddy currents. Quart. Appl. Math., 44(4):675–690, 1987. https://doi.org/10.1090/qam/872820
F. Paronetto. Existence results for a class of evolution equations of mixed type. J. Funct. Anal., 212(2):324–356, 2004. https://doi.org/10.1016/j.jfa.2004.03.014
F. Paronetto. Homogenization of degenerate elliptic-parabolic equations. Asymptot. Anal., 37(1):21–56, 2004.
V. Pluschke. Solution of a quasilinear parabolic-elliptic boundary value problem. In Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), volume 215 of Lecture Notes in Pure and Appl. Math., pp. 265–276. Dekker, New York, 2001. https://doi.org/10.1201/9780429187810-22
A. Quarteroni and A. Valli. Numerical approximation of partial differential equations, volume 23 of Springer Series in Computational Mathematics. SpringerVerlag, Berlin, 1994. https://doi.org/10.1007/978-3-540-85268-1
W. Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
R.E. Showalter. Degenerate evolution equations and applications. Indiana Univ. Math. J., 23:655–677, 1975. https://doi.org/10.1512/iumj.1974.23.23056
R.E. Showalter. Monotone operators in Banach space and nonlinear partial differential equations, volume 49 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997.
V. Thomée. Galerkin finite element methods for parabolic problems, volume 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006.
A. Ženíšek. Nonlinear elliptic and evolution problems and their finite element approximations. Computational Mathematics and Applications. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1990. With a foreword by P.-A. Raviart
E. Zeidler. Nonlinear functional analysis and its applications. II/A. SpringerVerlag, New York, 1990. Linear monotone operators, Translated from the German by the author and Leo F. Boron.
M. Zlámal. Finite element solution of quasistationary nonlinear magnetic field. RAIRO Anal. Numér., 16(2):161–191, 1982. https://doi.org/10.1051/m2an/1982160201611
M. Zlámal. Addendum to the paper: “Finite element solution of quasistationary nonlinear magnetic field” [RAIRO Anal. Numér. 16 (1982), no. 2, 161– 191; MR0661454 (83k:65086)]. RAIRO Anal. Numér., 17(4):407–415, 1983. https://doi.org/10.1051/m2an/1983170404071