Partial Differential Equations - Modelling and ... - ResearchGate

168 B. Maury
1
ε Ψ(x) −→ I K(x) as ε goes to 0, ∀x ∈ X.
If J ε admits a minimum u ε , for any ε, one can expect u ε to approach a (or
the) minimum of J over K, if it exists.
In actual Finite Element computations, some u ε h
is computed as the solution
to a finite dimensional problem, where h is a space-discretization parameter.
The present work is motivated by the fact that, even if the penalty
method for the continuous problem is convergent and the discretization procedure
is sound, the rate of convergence of u ε h
toward the exact solution is not
straightforward to obtain.
To our knowledge, the first paper dedicated to the analysis of the penalty
method in the Finite Element context dates back to 1973 (see [Bab73]), where
this method was used to incorporate Dirichlet boundary conditions in some
variants of the Finite Element Method. Since then, this strategy has been
followed to integrate obstacles in fluid flow simulations [ABF99], to model the
rigidity constraint [JLM05].
The present work is motivated by the handling of rigid particles in a
fluid flow. Various approaches have been proposed to incorporate rigid bodies
in a Stokes or Navier–Stokes fluid: arbitrary Lagrangian Eulerian approach
[JT96, Mau99], fictitious domain approach [PG02]. More recently, a strategy
based on augmented Lagrangian principles was proposed to handle a large
class of multimaterial flows [VCLR04, RPVC05]. In [JLM05], we tested the
raw penalty method to handle the rigidity constraint in a viscous fluid. This
approach is not sophisticated: it simply consists in adding to the variational
formulation the term
∫
1 (
∇u + ∇ T u ) : ( ∇v + ∇ T v ) .
ε Ω
It presents some drawbacks: as it is based on pure penalty and not augmented
Lagrangian, the penalty parameter has to be taken very small for the
constraint to be fulfilled properly, which may harm the conditioning of the
system to solve. Yet, it shows itself to be robust in practice, it allows the use
of non-boundary-fitted (e.g., Cartesian) meshes. Besides, it is straightforward
to implement, so that a full Navier–Stokes solver (in 2D) with circular rigid
particles can be written in about 50 lines, by using, for example, FreeFem++
(created by O. Pironneau, see [FFp]). Note that new tools for 3D problems
are already available (see, e.g., [ff3, DPP03] or [lif]), which enable to perform
computations of three dimensional fluid-particle flows.
We shall actually focus here on a simpler problem (see the problem (8)),
which is a scalar version of the rigidity constraint. The fluid velocity is indeed
replaced by a temperature field, and the rigid particle is replaced by a zone
with infinite conductivity. The Lagrange multiplier can be interpreted in this
context as a heat source term (see Remark 6).
We begin by presenting some standard properties of the penalty method
for quadratic optimization (Section 2.1), and some convergence results. Then