In this paper, we consider the extension of the finite element exterior calculus from elliptic problems, in which the Hodge Laplacian is an appropriate model problem, to parabolic problems, for which we take the Hodge heat equation as our model problem. The numerical method we study is a Galerkin method based on a mixed variational formulation and using as subspaces the same spaces of finite element differential forms that are used for elliptic problems. We analyze both the semidiscrete and a fully-discrete numerical scheme.
Accepté le :
DOI : 10.1051/m2an/2016013
Mots-clés : Finite element exterior calculus, mixed finite element method, parabolic equation, Hodge heat equation
@article{M2AN_2017__51_1_17_0,
author = {Arnold, Douglas N. and Chen, Hongtao},
title = {Finite element exterior calculus for parabolic problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {17--34},
publisher = {EDP-Sciences},
volume = {51},
number = {1},
year = {2017},
doi = {10.1051/m2an/2016013},
mrnumber = {3600999},
zbl = {1362.65100},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016013/}
}
TY - JOUR AU - Arnold, Douglas N. AU - Chen, Hongtao TI - Finite element exterior calculus for parabolic problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 17 EP - 34 VL - 51 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016013/ DO - 10.1051/m2an/2016013 LA - en ID - M2AN_2017__51_1_17_0 ER -
%0 Journal Article %A Arnold, Douglas N. %A Chen, Hongtao %T Finite element exterior calculus for parabolic problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 17-34 %V 51 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016013/ %R 10.1051/m2an/2016013 %G en %F M2AN_2017__51_1_17_0
Arnold, Douglas N.; Chen, Hongtao. Finite element exterior calculus for parabolic problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 17-34. doi : 10.1051/m2an/2016013. http://archive.numdam.org/articles/10.1051/m2an/2016013/
A. Alonso Rodríguez and A. Valli, Eddy current approximation of Maxwell equations. Theory, algorithms and applications. Vol. 4 of MS&A Modeling, Simulation and Applications. Springer-Verlag Italia, Milan (2010). | MR | Zbl
, and , Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1–155. | DOI | MR | Zbl
, and , Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.) 47 (2010) 281–354. | DOI | MR | Zbl
and , Analysis of finite element approximation of evolution problems in mixed form. SIAM J. Numer. Anal. 42 (2004) 1502–1526. | DOI | MR | Zbl
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011). | MR | Zbl
, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), 129–151. | Numdam | MR | Zbl
Th. Cazenave and A. Haraux, An introduction to semilinear evolution equations. Vol. 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York (1998). | MR | Zbl
, and , Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data. Numer. Math. 78 (1998) 495–521. | DOI | MR | Zbl
and , Error estimates for mixed finite element methods for nonlinear parabolic problems. Numer. Methods Partial Differ. Eq. 8 (1992) 395–404. | DOI | MR | Zbl
and , Smoothed projections in finite element exterior calculus. Math. Comput. 77 (2008) 813–829. | DOI | MR | Zbl
and , Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 7 (1970) 575–626. | DOI | MR | Zbl
, Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: the continuous-time case. Numer. Methods Partial Differ. Eq. 10 (1994) 129–147. | DOI | MR | Zbl
, Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: the discrete-time case. Numer. Methods Partial Differ. Eq. 10 (1994) 149–169. | DOI | MR | Zbl
A. Gillette, M. Holst and Y. Zhu, Finite element exterior calculus for evolution problems. Preprint (2015). | arXiv | MR
and , Generalization of the Ginzburg–Landau equations for non-stationary problems in the case of alloys with paramagnetic impurities. Soviet Phys. J. Exp. Theor. Phys. 27 (1968) 328–334.
M. Holst and Ch. Tiee, Finite element exterior calculus for parabolic evolution problems on riemannian hypersurfaces. Preprint (2015). | arXiv | MR
and , Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numér. 15 (1981) 41–78. | DOI | Numdam | MR | Zbl
A. Logg, K.-A. Mardal, G.N. Wells, et al., Automated solution of differential equations by the finite element method. Springer (2012). | Zbl
, Optimal -estimates for a mixed finite element method for second order elliptic and parabolic problems. Calcolo 20 (1983) 355–377. | DOI | MR | Zbl
, Superconvergence of mixed finite element methods for parabolic equations. RAIRO Modél. Math. Anal. Numér. 21 (1987) 327–352. | DOI | Numdam | MR | Zbl
V. Thomée, Galerkin finite element methods for parabolic problems, 2nd edition. Vol. 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2006). | MR
, A priori error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723–759. | DOI | MR | Zbl
, Differential forms, fluids, and finite models. Proc. Amer. Math. Soc. 139 (2011) 2597–2604. | DOI | MR | Zbl
Cité par Sources :
