We present a second order time-stepping scheme for parabolic problems on moving domains and interfaces. The diffusion coefficient is discontinuous and jumps across an interior interface. This causes the solution to have discontinuous derivatives in space and time. Without special treatment of the interface, both spatial and temporal discretization will be sub-optimal. For such problems, we develop a time-stepping method, based on a cG(1) Eulerian space-time Galerkin approach. We show −both analytically and numerically− second order convergence in time. Key to gaining the optimal order of convergence is the use of space-time test- and trial-functions, that are aligned with the moving interface. Possible applications are multiphase flow or fluid-structure interaction problems.
Accepté le :
DOI : 10.1051/m2an/2016072
Mots clés : Space-time finite elements, time stepping, moving interfaces, a priori error analysis
@article{M2AN_2017__51_4_1539_0,
author = {Frei, Stefan and Richter, Thomas},
title = {A second order time-stepping scheme for parabolic interface problems with moving interfaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1539--1560},
publisher = {EDP-Sciences},
volume = {51},
number = {4},
year = {2017},
doi = {10.1051/m2an/2016072},
mrnumber = {3702424},
zbl = {1379.65076},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016072/}
}
TY - JOUR AU - Frei, Stefan AU - Richter, Thomas TI - A second order time-stepping scheme for parabolic interface problems with moving interfaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1539 EP - 1560 VL - 51 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016072/ DO - 10.1051/m2an/2016072 LA - en ID - M2AN_2017__51_4_1539_0 ER -
%0 Journal Article %A Frei, Stefan %A Richter, Thomas %T A second order time-stepping scheme for parabolic interface problems with moving interfaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1539-1560 %V 51 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016072/ %R 10.1051/m2an/2016072 %G en %F M2AN_2017__51_4_1539_0
Frei, Stefan; Richter, Thomas. A second order time-stepping scheme for parabolic interface problems with moving interfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1539-1560. doi : 10.1051/m2an/2016072. http://www.numdam.org/articles/10.1051/m2an/2016072/
and , Continuous finite elements in space and time for the heat equation. Math. Comput. 52 (1989) 255–274. | DOI | MR | Zbl
, The finite element method for elliptic equations with discontinuous coefficients. Computing 5 (1970) 207–213. | DOI | MR | Zbl
and , Fully implicit time discretization for a free surface flow problem. Proc. Appl. Math. Mech. 11 (2011) 619–620. | DOI
and , An unfitted finite element method using discontinuous Galerkin, Int. J. Numer. Methods Eng. 79 (2009) 1557–1576. | DOI | MR | Zbl
and , A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6 (1996) 109–138. | DOI | MR | Zbl
, , and , The fixed-mesh ale approach for the numerical approximation of flows in moving domains. J. Comput. Phys. 228 (2009) 1591–1611. | DOI | MR | Zbl
G.-H. Cottet, E. Maitre and T. Milcent, An Eulerian method for fluid-structure coupling with biophysical applications, in ECCOMAS CFD 2006: Proc. of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands (2006).
T. Dunne and R. Rannacher, Adaptive finite element approximation of fluid-structure interaction based on an Eulerian variational formulation, in Fluid-structure interaction. Springer (2006) 110–145. | MR
R. Dziri and J.-P. Zolésio, Eulerian derivative for non-cylindrical functionals, in Vol. 216 of Shape Optimization and Optimal Design, edited by J. Cagnol, M.P. Polis and J.-P. Zolésio (2001) 87–108. | MR | Zbl
K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational differential equations. Cambridge Univ. Press (1996). | MR | Zbl
S. Frei, Eulerian finite element methods for interface problems and fluid-structure interactions. Ph.D. thesis, Universität Heidelberg (2016).
and , A locally modified parametric finite element method for interface problems. SIAM J. Numer. Anal. 52 (2014) 2315–2334. | DOI | MR | Zbl
and , On time integration in the XFEM. Int. J. Numer. Methods Eng. 79 (2009) 69–93. | DOI | MR | Zbl
and , Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. | DOI | MR | Zbl
and , Analysis of a Nitsche XFEM DG discretization for a class of two-phase mass transport problems. SIAM J. Numer. Anal. 51 (2013) 958–983. | DOI | MR | Zbl
, and , A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46 (1999) 131–150. | DOI | MR | Zbl
T. Richter, Finite Elements for Fluid-Structure Interactions. Springer (2016), in preparation. | MR
and , Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods. Comput. Methods Appl. Mech. Eng. 258 (2013) 39–54. | DOI | MR | Zbl
, Analysis of backward Euler/extended finite element discretization of parabolic problems with moving interfaces. Comput. Methods Appl. Mech. Eng. 258 (2013) 152–165. | DOI | MR | Zbl
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