Partial Differential Equations - Modelling and ... - ResearchGate

184 B. Maury
The source term g is, therefore, a single layer distribution supported by ∂O,
with weight −λ · n = −∂u/∂n (where n is the outward normal to ∂O). Note
that it is in H 1/2 (∂O).
Remark 7. Note that letting ε go to 0 for any h>0 leads to an estimate for
a fictitious domain method (à laGlowinski, i.e. based on the use of Lagrange
multiplier). In [GG95], an error estimate is obtained for such a method; it
relies on two independent meshes for the primal and dual components of the
solution (conditionally to some compatibility conditions between the sizes of
the two meshes). We recover this estimate in the situation where the local
mesh is simply the restriction of the covering mesh to the obstacle (to the
reduced obstacle O h , to be more precise).
Remark 8. The approach we presented can be extended to other situations,
like the one we already considered in Example 2, as soon as a H 1 -penalty is
used. The functional to minimize is then
J ε (v) = 1 ∫ ∫
|∇v| 2 − fv + 1 ∫ (
u 2 + |∇u| 2) ,
2
2ε
Ω
Ω
so that B identifies to the restriction operator from H 1 0 (Ω) toH 1 (O). The discrete
inf-sup condition, as well as the approximation properties, are essentially
the same as in the case we considered here.
Concerning the original problem of simulating fluid-particle flows, an extra
difficulty lies in the fact that two constraints of different types must be dealt
with (global incompressibility and local rigid motion). It raises additional
issues which shall be addressed in the future.
Ω
References
[ABF99] Ph. Angot, Ch.-H. Bruneau, and P. Fabrie. A penalization method
to take into account obstacles in incompressible viscous flows. Numer.
Math., 81(4):497–520, 1999.
[AR] Ph. Angot and I. Ramière. Convergence analysis of the Q1-finite element
method for elliptic problems with non boundary-fitted meshes.
Submitted.
[Bab73] I. Babuška. The finite element method with penalty. Math. Comp.,
27:221–228, 1973.
[BF91] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods,
volume 15 of Springer Series in Computational Mathematics. Springer-
Verlag, New York, 1991.
[DP02] S. Del Pino. Une méthode d’éléments finis pour la résolution d’EDP
dans des domaines décrits par géométrie constructive. PhD thesis,
Université Pierre et Marie Curie, Paris, 2002.
[DPP03] S. Del Pino and O. Pironneau. A fictitious domain based general PDE
solver. In E. Heikkola, editor, Numerical Methods for Scientific Computing,
Barcelona, 2003.