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/
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