Surface problems play a key role in several theoretical and applied fields. In this work the main focus is the presentation of a detailed analysis of the approximation of the classical porous media flow problem: the Darcy equation, where the domain is a regular surface. The formulation considers the mixed form and the numerical approximation adopts a classical pair of finite element spaces: piecewise constant for the scalar fields and Raviart–Thomas for vector fields, both written on the tangential space of the surface. The main result is the proof of the order of convergence where the discretization error, due to the finite element approximation, is coupled with a geometrical error. The latter takes into account the approximation of the real surface with a discretized one. Several examples are presented to show the correctness of the analysis, including surfaces with boundary.
Accepté le :
DOI : 10.1051/m2an/2015095
Mots clés : PDEs on surfaces, Darcy problem, mixed finite elements
@article{M2AN_2016__50_6_1615_0,
author = {Ferroni, Alberto and Formaggia, Luca and Fumagalli, Alessio},
title = {Numerical analysis of {Darcy} problem on surfaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1615--1630},
publisher = {EDP-Sciences},
volume = {50},
number = {6},
year = {2016},
doi = {10.1051/m2an/2015095},
mrnumber = {3580116},
zbl = {1457.65195},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015095/}
}
TY - JOUR AU - Ferroni, Alberto AU - Formaggia, Luca AU - Fumagalli, Alessio TI - Numerical analysis of Darcy problem on surfaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1615 EP - 1630 VL - 50 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015095/ DO - 10.1051/m2an/2015095 LA - en ID - M2AN_2016__50_6_1615_0 ER -
%0 Journal Article %A Ferroni, Alberto %A Formaggia, Luca %A Fumagalli, Alessio %T Numerical analysis of Darcy problem on surfaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1615-1630 %V 50 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015095/ %R 10.1051/m2an/2015095 %G en %F M2AN_2016__50_6_1615_0
Ferroni, Alberto; Formaggia, Luca; Fumagalli, Alessio. Numerical analysis of Darcy problem on surfaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1615-1630. doi : 10.1051/m2an/2015095. http://www.numdam.org/articles/10.1051/m2an/2015095/
, , , , and , High order discontinuous Galerkin methods on surfaces. SIAM J. Numer. Anal. 53 (2015) 1145–1171. | DOI | MR | Zbl
J. Bear, Dynamics of Fluids in Porous Media. American Elsevier (1972). | Zbl
, , and , Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174 (2001) 759–780. | DOI | MR | Zbl
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Vol. 15 of Comput. Math. Springer Verlag, Berlin (1991). | Zbl
, , , and , High-order spectral/ element discretisation for reaction-diffusion problems on surfaces: Application to cardiac electrophysiology. J. Comput. Phys. 257 (2014) 813–829. | DOI | Zbl
P.G. Ciarlet, Mathematical Elasticity Volume I: Three-Dimensional Elasticity. Vol. 20 of Stud. Math. Appl. Elsevier (1988). | Zbl
and , A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. Math. Model. Numer. Anal. 46 (2012) 465–489. | DOI | Numdam | Zbl
, , and , Efficient geometric reconstruction of complex geological structures. Math. Comput. Simul. 106 (2014) 163–184. | DOI | Zbl
M.C. Delfour and J.-P. Zolésio, Shapes and Geometries, 2nd edition. Society for Industrial and Applied Mathematics (2011). | Zbl
and , An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45 (2007) 421–442. | DOI | Zbl
G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces. In Partial Differential Equations and Calculus of Variations. Edited by S. Hildebrandt and R. Leis. Vol. 1357 of Lect. Notes Math. Springer, Berlin, Heidelberg (1988) 142–155. | Zbl
, and , Finite elements on evolving surfaces. IMA J. Numer. Anal. 27 (2007) 262–292. | DOI | Zbl
G. Dziuk and C.M Elliott, Surface finite elements for parabolic equations. J. Comput. Math. 25 (2007).
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer (2004). | Zbl
, , and , A reduced model for Darcy’s problem in networks of fractures. ESAIM: M2AN 48 (2014) 1089–1116. | DOI | Numdam | Zbl
G. Fourestey and S. Deparis, Lifev user manual. Available at http://lifev.org November (2010).
and , A numerical method for two-phase flow in fractured porous media with non-matching grids. Computational Methods in Geologic CO2 Sequestration. Adv. Water Resour. 62 (2013) 454–464. | DOI
and , An efficient XFEM approximation of Darcy flow in arbitrarly fractured porous media. Oil Gas Sci. Technol. – Revue d’IFP Energies Nouvelles 69 (2014) 555–564. | DOI
and , Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Found. Comput. Math. 12 (2012) 263–293. | DOI | Zbl
, and , Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667–1691. | DOI | Zbl
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Vol. 23 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1994). | Zbl
, and , Efficient assembly of h(div) and h(curl) conforming finite elements. SIAM J. Sci. Comput. 31 (2009) 4130–4151. | DOI | Zbl
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