Variational formulation for a nonlinear elliptic equation in a three-dimensional exterior domain

An existence result was obtained for a nonlinear second-order equation in an exterior domain of IR(3). The proof relies on a variational formulation in weighted Sobolev spaces

Continuity properties of the inf-sup constant for the divergence

The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant that can be obtained from the numerical solution of an eigenvalue problem for the Stokes system. This eigenvalue problem does not fall into a class for which standard results about numerical approximations can be applied. Indeed, many reasonable finite element methods do not yield a convergent approximation. In this article, we show that under fairly weak conditions on the approximation of the domain, the LBB constant is an upper semi-continuous shape functional, and we give more restrictive sufficient conditions for its continuity with respect to the domain. For numerical approximations based on variational formulations of the Stokes eigenvalue problem, we also show upper semi-continuity under weak approximation properties, and we give stronger conditions that are sufficient for convergence of the discrete LBB constant towards the continuous LBB constant. Numerical examples show that our conditions are, while not quite optimal, not very far from necessary

Analysis of an Electroless Plating Problem

Electroless plating in microfluidic channels is a novel technology at the micrometer scale. As the microchannel depth varies with the flow of the chemicals, care must be taken for the channel not to run dry. Owing to the deposited chemical species the physical domain of the flow changes with time, leading to a free boundary problem. As the motion of the free boundary is small it is modeled by a transpiration approximation. With this simplification, the mathematical model, consists of a Navier-Stokes flow and an equation for the concentration of the plating chemical coupled by non standard and nonlinear boundary conditions. Existence and uniqueness are proven for the concentration equation. Numerical analysis is carried out and justifies the proposed numerical schemes and nonlinear algorithms. A numerical study is performed, in the two dimensional case, with the finite element method and an implicit Euler time-scheme

On Friedrichs constant and Horgan-Payne angle for LBB condition

In dimension 2, the Horgan-Payne angle serves to construct a lower bound for the inf-sup constant of the divergence arising in the so-called LBB condition. This lower bound is equivalent to an upper bound for the Friedrichs constant. Explicit upper bounds for the latter constant can be found using a polar parametrization of the boundary. Revisiting carefully the original paper which establishes this strategy, we found out that some proofs need clarification, and some statements, replacement

Un résultat de trace pour les éléments finis de Crouzeix–Raviart, application à la discrétisation des équations de Darcy

Crouzeix-­Raviart finite elements give rise to a space of discontinuous functions which are affine on each element of a triangulation of the domain. The aim of this note is to prove a trace result for this space. We present an application to the discretization of Darcy's equations

Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell models. This confirms an earlier observation of Tanner. We provide a new interpretation of the temporal instability of the grade-two fluid model for negative coefficients. Our techniques allow a simple proof of the convergence of the steady grade-two model to the Navier–Stokes model as $\alpha \rightarrow 0$ (under suitable conditions) in three dimensions. They also provide a proof of the convergence of the steady Oldroyd models to the Navier–Stokes model as their parameters tend to zero

Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell models. This confirms an earlier observation of Tanner. We provide a new interpretation of the temporal instability of the grade-two fluid model for negative coefficients. Our techniques allow a simple proof of the convergence of the steady grade-two model to the Navier–Stokes model as $\alpha \rightarrow 0$ (under suitable conditions) in three dimensions. They also provide a proof of the convergence of the steady Oldroyd models to the Navier–Stokes model as their parameters tend to zero