Résumé
We propose a new reduced-order fictitious domain method for the discretization of the motion of small rigid particles immersed in a viscous incompressible flow. The reduced model is based on the projection of Dirichlet boundary constraints onto a finite-dimensional approximation space of Fourier type, thereby yielding a problem with defective interface conditions imposed through a Lagrange multiplier formulation. We analyze the existence of solutions to the reduced Stokes problem and, for arbitrarily small particles and prescribed rigid velocities at the boundary, we prove its convergence toward the full Stokes problem. The convergence rate depends on both the size of the inclusion and the number of modes in the finite-dimensional space. The numerical discretization of the reduced problem is carried out using the finite element method on a computational mesh that does not conform to the particles, within a fictitious domain framework. We propose a stabilized and robust formulation with respect to the particle size, and prove its stability and convergence in velocity, pressure, and interface fluid stress. A key step in the analysis is the derivation of an inf-sup condition involving both the divergence constraint and the interfacial Dirichlet constraints, with particular attention paid to the dependence on the particle size. The properties of the discretization method are further illustrated by numerical experiments.</p></div> <div>Weak formulations<p>In this section, we first introduce a mixed formulation of the coupled problem (1.5), (1.6) and (1.8) and show that it can be expressed, in an equivalent way, as a fictitious domain model formulation, in which the fluid equations are extended to the entire domain Ω, as the solid motions are rigid ones. Both the kinematic and dynamic coupling conditions between the particle and the fluid on the interface Bωεptq are enforced through Lagrange multipliers. In this formulation, referred to as full coupled problem, a new unknown is thus introduced, which is the Lagrange multiplier λε, defined on the interface Bωεptq, associated to the kinematic constraint (1.8)2. Finally, a reduced formulation is introduced where functional space of the Lagrange multiplier is approximated by a finite dimensional space, leading to an approximated problem, referred in the following as reduced coupled problem.</p><p>In what follows, we denote by L 2 0 pOq the closed subspace of L 2 pOq of functions with zero average on O. The scalar product on L 2 pOq is denoted by p¨, ¨qO and the associated norm by } ¨}O. The norm of the Sobolev space H s pOq, s ą 0 is denoted by } ¨}s,O. We also use the notation H ´1 2 pBOq to denote the dual space of H 1 2 pBOq, with associated duality pairing x¨, ¨y´1 2 , 1 2 BO . In the case of vector-valued functions, the associated Hilbert spaces will be indicated in boldface.</p></div> <div>Full coupled problem<p>We consider the following mixed weak formulation of the full coupled problem (1.5), (1.6) and ( 1.8): Find puε, pε, λε, rε, θεq such that puεptq, pεptq, λεptq, rεptq, θεptqq P H 1 BΩ `Ωεptq ˘ˆL 2 0 pΩεptqq ˆH´1 2 pBωεptqq ˆR2 ˆR a.e. t P R ànd ϕεptq " rεptq `ΛεptqI, ωεptq " ϕεpp ωε, tq, Ωεptq " Ωzωεptq, $ & % a f Ωεptq `puε, pεq, pv, qq ˘`ρ b A ε b : rε ¨δr `ρb I ε b : θεδθ ´