Efficiency of approximate boundary conditions for corner domains coated with thin layers

In this Note, we consider an interface problem posed in a bounded domain with thin layer. In the case of a smooth domain, approximate boundary conditions (also called impedance conditions) are known to approximate in a precise way the effect of the layer, as its thickness goes to zero. We investigate here the efficiency of such conditions when the domain has a corner; we show that it deteriorates when the opening of the corner angle grows, giving optimal estimates thanks to multiscale asymptotic expansions. Numerical results are given, which illustrate these estimates. To cite this article: G. Vial, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Solution of a system of nonstrictly hyperbolic conservation laws

In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL ^∞ functions always but may contain δ-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution.     

Generalized impedance boundary condition at high frequency for a domain with thin layer: the circular case

Consider a conducting disk surrounded by a thin dielectric layer submitted to an electric field at the pulsation ω. The conductivity of the layer grows like ω^1?γ, γ∈[0,1], when the pulsation ω?tends to infinity. Using a pseudodifferential approach on the torus, we build an equivalent boundary condition with the help of an appropriate factorization of Helmholtz operator in the layer. This generalized impedance condition approximates the thin membrane in the high frequency limit for small thickness of the layer. L ²-error estimates are given and we illustrate our results with numerical simulations. This work extends, in the circular geometry, previous works of Lafitte and Lebeau (Lafitte O. Lebeau G. 1993, Équations de Maxwell et opérateur d’impédance sur le bord d’un obstacle convexe absorbant. Comptes Rendus de l ' Académic dis Science, Paris, Série I, Mathématiques, 316(11), 1177–1182); (Lafitte O.D., 1999, Diffraction in the high frequency regime by a thin layer of dielectric material. I. The equivalent impedance boundary condition. SIAM Journal on Applied Mathematics, 59(3), 1028–1052 (electronic)) in which γ?identically equals zero.     

Study of a perforated thin plate according to the relative sizes of its different parameters

We are interested in the study of a thin plate, periodicially perforated by cylindrical holes, the axes of which are perpendicular to the plane of the plate. A horizontal section of the plate specifies its geometry, and shows a periodicity in the order of ?. The thickness of the plate is equal to e. The ratio of material is small, and is characterized by the parameter δ, the thickness of the bars being equal to ?δ. In this paper, we study the dependence of displacements on e, ? and δ, and to give equivalent limits when e, then ?, and finally δ, tend towards zero. An interesting result obtained in this work is the negative Poisson coefficient of the final equivalent material. Although this coefficient is theoretically between ?½ and 1, most materials encountered in practice have a positive one.     

Optimal control of unilateral obstacle problem with a source term

We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in H². We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions existence and give necessary optimality conditions. The author is grateful to Prof. M. Bergounioux for her instructive suggestions.     

Compressing thin spheres in the complement of a link

Let L be a link in S³ that is in thin position but not in bridge position and let P be a thin level sphere with compressing disk D. We introduce the idea of alternating level spheres for D and show that all such spheres are thin and their widths are monotone decreasing. This allows us to generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks P can have in terms of the width of P, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on some side or be weakly incompressible. In particular we show that the thin level sphere of second lowest width is weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.     

Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler

We construct asymptotics for the eigenvalues and vector eigenfunctions of the elasticity problem for an anisotropic body with a thin coupler (of diameter h) attached to its surface. In the spectrum we select two series of eigenvalues with stable asymptotics. The first series is formed by eigenvalues O(h ²) corresponding to the transverse oscillations of the rod with rigidly fixed ends, while the second is generated by the longitudinal oscillations and twisting of the rod, as well as eigenoscillations of the body without the coupler. We check the convergence theorem for the first series and derive the error estimates for both series.     

Existence of solutions for a nonlocal epitaxial thin film growing equation

The existence of weak solutions is studied to the initial boundary problem of a nonlocal epitaxial thin film growing equation modeling epitaxial thin film growth. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.     

Solution of heat-conduction problems for thin shells and plates by application of a generalized variational approach

By applying a generalized variational approach we construct an approximate system of equations for heat conduction for thin shells and plates and develop a method of solving them. We give the results of numerical studies for a particular problem.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 66–70.     

Transmission problems in a thin layer set in the framework of the Hölder spaces: Resolution and impedance concept

We consider a family of singular transmission problems depending on some small positive parameter δ set in the juxtaposition of two rectangular domains. They are written in the form of abstract elliptic equations and the study, here, is given in the Hölder spaces completing in this way the work in Lp cases given in [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut. 18 (1) (2005) 143-176]. In this first part, we present a new approach for the resolution of these problems by using the concept of impedance operator. This method is different of the one performing a rescaling in the thin layer, see [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut. 18 (1) (2005) 143-176]. It leads to obtain direct and simplified problems. We use the Dunford calculus and some techniques similar to that in [R. Labbas, Problèmes aux limites pour une équation différentielle abstraite de type elliptique, Thèse d'état, Université de Nice, 1987; A. Favini, R. Labbas, S. Maingot, H. Tanabe, A. Yagi, Unified study of elliptic problems in Hölder spaces, C. R. Math. Acad. Sci. Paris 134 (2005); G. Dore, A. Favini, R. Labbas, K. Lemrabet, S. Maingot, A transmission problem in a thin layer, Part I, Sharp estimates, in press], to prove existence, uniqueness, results and some specific estimates on the impedance operator. This study will allow us, in a forthcoming work, to obtain respectively optimal regularities and the limit problem when δ→0.     

Numerical solution of a minimax ergodic optimal control problem

In this work we consider an L^∞ minimax ergodic optimal control problem with cumulative cost. We approximate the cost function as a limit of evolutions problems. We present the associated Hamilton-Jacobi-Bellman equation and we prove that it has a unique solution in the viscosity sense. As this HJB equation is consistent with a numerical procedure, we use this discretization to obtain a procedure for the primitive problem. For the numerical solution of the ergodic version we need a perturbation of the instantaneous cost function. We give an appropriate selection of the discretization and penalization parameters to obtain discrete solutions that converge to the optimal cost. We present numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element approximations to surfaces of prescribed variable mean curvature

We give an algorithm for finding finite element approximations to surfaces of prescribed variable mean curvature, which span a given boundary curve. We work in the parametric setting and prove optimal estimates in the H¹ norm. The estimates are verified computationally.     

Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.     

Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer

We study the behaviour of steady‐state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively, α) surrounded by a thin membrane of thickness h and of complex permittivity α (respectively, 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady‐state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter α is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady‐state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so‐called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and α with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviours of the potentials. Copyright © 2007 John Wiley & Sons, Ltd.     

On the mixed finite element approximation for the buckling of plates

We consider the mixed finite element method for the buckling problem of the thin plate by using piecewise linear polynomials. We give error estimates for the approximate eigenvalues and the eigenfunctions.     

Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems

We study the numerical approximation of distributed optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the L ²-error estimates are of order o(h), which is optimal according with the $C^{0,1}(\overline{\Omega})$ -regularity of the optimal control.     

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

We consider a finite element method (FEM) with arbitrary polynomial degree for nonlinear monotone elliptic problems. Using a linear elliptic projection, we first give a new short proof of the optimal convergence rate of the FEM in the L² norm. We then derive optimal a priori error estimates in the H¹ and L² norm for a FEM with variational crimes due to numerical integration. As an application, we derive a priori error estimates for a numerical homogenization method applied to nonlinear monotone elliptic problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 955–969, 2016     

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

In this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L^∞(L²) for velocity and concentration and the super convergence in L^∞(H¹) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.     

An asymptotic result on principal points for univariate distributions

We give an asymptotic result for principal points of univariate distributions, as defined in Flury (1990). Principal points are a generalization of the mean and provide a natural way to approximate a continuous distribution. We show that for a given density $si:f$esi:it is asymptotically optimal to take the quantiles of the density proportional to$si:f$esi:i^1/3 as principal points.     

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is O(h ²) or \(O\left( {{h^2}\sqrt {\left| {\ln h} \right|} } \right)\) in the sense of L ²-norm or L ^∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.