Ventcel Boundary Conditions Research Articles

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

<p style='text-indent:20px;'>This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.</p>

Controllability of a Stokes system with a diffusive boundary condition

We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.

Theoretical justification of Ventcel's boundary conditions for a thin layer three-dimensional thermoelasticity problem

Theoretical justification of Ventcel's boundary conditions for a thin layer three-dimensional thermoelasticity problem

Second-Order Differential Operators with Non-Local Ventcel's Boundary Conditions

Different boundary conditions have been introduced for second-order differential operators and the properties of the operators on the corresponding domains have been deeply investigated since the work of Feller. The aim of this paper is to study second-order differential operators satisfying a Ventcel's type boundary condition which involves simultaneously both the endpoints of a real interval. We study different general properties and a resolvent estimate for this kind of operators.

Stabilization of the wave equation with Ventcel boundary condition

We consider a damped wave equation on a open subset of Rn or a smooth Riemannian manifold with boundary, with Ventcel boundary conditions, with a linear damping, acting either in the interior or at the boundary. This equation is a model for a vibrating structure with a layer with higher rigidity of thickness δ>0. By means of a proper Carleman estimate for second-order elliptic operators near the boundary, we derive a resolvent estimate for the wave semigroup generator along the imaginary axis, which in turn yields the logarithmic decay rate of the energy. This stabilization result is obtained uniformly in δ.

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the finite element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.

Reinforcement of a Mindlin–Timoshenko plate by a thin layer

We consider a problem of reinforcement of a Mindlin–Timoshenko plate with a thin stiffener of thickness \({\delta }\), on a portion of its boundary. We investigate the case where the mass density, the rigidity and the shear modulus of the material constituting the stiffener vary as \({\delta ^{-a}}\), where \({a\in \mathbb{R_{+}^{\ast }}}\). We perform an asymptotic analysis of the solution as \({\delta }\) goes to zero. We shall show that different limit behaviors occur when a vary in \({\mathbb{R_{+}^{\ast }}}\). The situation where the stiffener is of variable thickness is also investigated. It is also shown how the Kirchhoff–Love model, with Ventcel boundary conditions, is obtained, when the shear modulus approaches +\({\infty}\) (when it behaves as specific power of \({\delta}\)) in both the plate and the stiffener.

Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

A semilinear parabolic transmission problem with Ventcel's boundary conditions on a fractal interface S or the corresponding prefractal interface Sh is studied. Regularity results for the solution in both cases are proved. The asymptotic behaviour of the solutions of the approximating problems to the solution of limit fractal problem is analyzed.

Shape optimization methods for the inverse obstacle problem with generalized impedance boundary conditions

We aim to reconstruct an inclusion ω immersed in a perfect fluid flowing in a larger bounded domain Ω via boundary measurements on ∂Ω. The obstacle ω is assumed to have a thin layer and is then modeled using generalized boundary conditions (precisely Ventcel boundary conditions). We first obtain an identifiability result (i.e. the uniqueness of the solution of the inverse problem) for annular configurations through explicit computations. Then, this inverse problem of reconstructing ω is studied, thanks to the tools of shape optimization by minimizing a least-squares-type cost functional. We prove the existence of the shape derivatives with respect to the domain ω and characterize the gradient of this cost functional in order to make a numerical resolution. We also characterize the shape Hessian and prove that this inverse obstacle problem is unstable in the following sense: the functional is degenerate for highly oscillating perturbations. Finally, we present some numerical simulations in order to confirm and extend our theoretical results.

Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coefficient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surfaces are sought for numerical purposes. The well-posedness of such problems was recently investigated: up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation.

Polynomial stabiization of the wave equation with Ventcel's boundary conditions

AbstractWe consider the wave equation on the unit square of the plane with Ventcel boundary conditions on a part of the boundary. It was shown by A. Heminna [8] that this problem is not exponentially stable. Here using a Fourier analysis and a careful analysis of the 1‐d problem with respect to the Fourier parameter l, we show a polynomial stability of this system (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On Generalized Ventcel's Type Boundary Conditions for Laplace Operator in a Bounded Domain

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of a coefficient. Nevertheless, situations where this condition is violated appeared in several works. The well-posedness of such problems was still open. This manuscript establishes, in the generic case, the existence and uniqueness of the solution for the Ventcel boundary value problem without the sign condition. Then we consider perforated geometries and give conditions to remove the genericity restriction.

Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions: $$\left\{\begin{array}{l@{\qquad}l@{\quad}l} u_{tt}- \Delta u +a(x)g(u_t)=0 \quad {\rm in}\quad \Omega\;\times ] 0,\infty[\\ \partial_{\nu} u - \Delta_{\Gamma_1}u=0 \qquad \qquad \quad {\rm on}\quad \Gamma_{1}\times ] 0,\infty[\\ u=0\qquad \qquad \qquad \quad \qquad {\rm on} \quad\Gamma_{0}\times ] 0,\infty [\end{array}\right.$$ where Ω is a bounded domain of \({\mathbb{R}^{n}}\) (n ≥ 2) having a smooth boundary \({\Gamma :=\partial \Omega}\) , such that \({ \Gamma =\Gamma _{0} \cup \Gamma_{1}}\) with \({\Gamma_0}\) , \({\Gamma_1}\) being closed and disjoint. It is known that if a(x) = 0 then the uniform exponential stability never holds even if a linear frictional feedback is applied to the entire boundary of the domain [see, for instance, Hemmina (ESAIM, Control Optim Calc Var 5:591–622, 2000, Thm. 3.1)]. Let \({f:\overline{\Omega} \rightarrow \mathbb R}\) be a smooth function; define ω1 to be a neighbourhood of \({\Gamma_1}\) , and subdivide the boundary \({\Gamma_0}\) into two parts: \({\Gamma_0^{\ast}=\{x\in \Gamma_0;\partial_{\nu}f > 0\}}\) and \({\Gamma_0 \backslash \Gamma_0^{\ast}}\) . Now, let ω0 be a neighbourhood of \({\overline{\Gamma_0^{\ast}}}\) . We prove that if a(x) ≥ a0 > 0 on the open subset \({\omega =\omega_0 \cup \omega_1}\) and if g is a monotone increasing function satisfying k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, then the energy of the system decays uniformly at the rate quantified by the solution to a certain nonlinear ODE dependent on the damping [as in Lasiecka and Tataru (Differ Integral Equ 6:507–533, 1993)].

Ventcel's boundary conditions and analytic semigroups

In this paper we prove that first-order degenerate ordinary differential operators generate analytic semigroups in spaces of continuous functions if Ventcel's boundary conditions are assumed. Some applications to different types of degeneration are also discussed.

On C0-semigroups generated by differential operators satisfying Ventcel's boundary conditions

On C0-semigroups generated by differential operators satisfying Ventcel's boundary conditions