Anisotropic Singular Perturbation Research Articles

Uniform estimates for conforming Galerkin method for anisotropic singularly perturbed elliptic problems

In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming Q 1 finite element method are derived, and some other results of convergence and regularity for the continuous problem are proved. The arguments are based on the tensorization trick and standard elliptic regularity theory tools.

On some new results on anisotropic singular perturbations of second-order elliptic operators

In this article, we deal with some problems involving a class of singularly perturbed elliptic operators. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin approximation to estimate the convergence rate on the whole domain, for the linear problem. Finally, we study the asymptotic behaviour of the semigroup generated.

Anisotropic non-local problems: asymptotic behaviour and existence results

ABSTRACT We deal with anisotropic singular perturbation problems in some weighted spaces of Sobolev type. The perturbed problems are elliptic, semi-linear and non-local. Using a variational approach, we establish the existence of their solutions as critical points of some -functionals. Besides, we study the asymptotic behaviour of these solutions with respect to the parameter of perturbation ε and, as , we obtain existence results for some non-standard integro-differential problems. A non-local eigenvalue problem related to the considered problems is also investigated and used to carry out this study.

W2,2 interior convergence for some class of elliptic anisotropic singular perturbations problems

ABSTRACTIn this paper, we deal with anisotropic singular perturbations of some class of elliptic problems. We study the asymptotic behavior of the solution in a certain second-order pseudo Sobolev space.

Anisotropic singular perturbations of variational inequalities

We consider variational inequalities involving the p-Laplace operator with anisotropic singular perturbations where the convex set, on which the problem is defined, is also subject to perturbations. This leads to introduce a new convergence of sets, in some suitable sense, conceived from the Mosco convergence and matching well to the anisotropic singular perturbations. Convergence results and their rates are established. In order to illustrate the introduced convergence sets, obstacle and elasto-plastic perturbed problems are dealt with. This allows to go deeper in the analysis of the suggested convergence on concrete sets in Sobolev spaces.

Asymptotic development of anisotropic singular perturbation problems

Asymptotic development of anisotropic singular perturbation problems

Existence Results for Some Partial Integro-Differential Equations

In this note, we deal with semilinear integro-differential equations subject to homogeneous Dirichlet boundary conditions given on the boundaries of the sections. Even if the differentiation will be taken only in some directions, it is not possible to see the main problem parameterized by the other coordinates because of the non-local terms which also obliged the problem to be degenerate. We establish the existence of solutions by employing the singular perturbations method as a natural tool. The perturbed problems are classical, non-local, semilinear elliptic problems and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. Some improvement, concerning the existence of the solutions and the convergence results depending on the weights, will be established. The paper also gives an idea about the study of the anisotropic singular perturbations in the framework of weighted spaces.

On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems

In this article we give an extention of the L^2-theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear problems involving L^p data (1<p<2). Convergences in pseudo Sobolev spaces are proved for weak and entropy solutions, and rate of convergence is given in cylindrical domains

On some anisotropic, nonlocal, parabolic singular perturbations problems

This article is devoted to the study of the anisotropic singular perturbations theory for some quasilinear parabolic problems. Describing the asymptotic behaviour of the solutions of nonlocal problems yields to show the existence of solution to some integro-differential problems. The closeness between the solutions of these two kinds of problems is established.

RIGOROUS DERIVATION OF A HYPERBOLIC MODEL FOR TAYLOR DISPERSION

In this paper, we upscale the classical convection-diffusion equation in a narrow slit. We suppose that the transport parameters are such that we are in Taylor's regime, i.e. we deal with dominant Péclet numbers. In contrast to the classical work of Taylor, we undertake a rigorous derivation of the upscaled hyperbolic dispersion equation. Hyperbolic effective models were proposed by several authors and our goal is to confirm rigorously the effective equations derived by Balakotaiah et al. in recent years using a formal Lyapounov–Schmidt reduction. Our analysis uses the Laplace transform in time and an anisotropic singular perturbation technique, the small characteristic parameter ε being the ratio between the thickness and the longitudinal observation length. The Péclet number is written as Cε-α, with α &lt; 2. Hyperbolic effective model corresponds to a high Péclet number close to the threshold value when Taylor's regime turns to turbulent mixing and we characterize it by assuming 4/3 &lt; α &lt; 2. We prove that the difference between the dimensionless physical concentration and the effective concentration, calculated using the hyperbolic upscaled model, divided by ε2-α (the local Péclet number) converges strongly to zero in L2-norm. For Péclet numbers considered in this paper, the hyperbolic dispersion equation turns out to give a better approximation than the classical parabolic Taylor model.

Anisotropic singular perturbations of hyperbolic problems

Anisotropic singular perturbations of linear hyperbolic problems are considered. A description of the asymptotic behaviour of the solution as ε → 0 is given. In the case of cylindrical domains, we improve the rate of convergence in regions far from the lateral boundary.

Correctors for some asymptotic problems

In the theory of anisotropic singular perturbation boundary value problems, the solution uɛ does not converge, in the H1-norm on the whole domain, towards some u0. In this paper we construct correctors to have good approximations of uɛ in the H1-norm on the whole domain. Since the anisotropic singular perturbation problems can be connected to the study of the asymptotic behaviour of problems defined in cylindrical domains becoming unbounded in some directions, we transpose our results for such problems.

Laplace transform approach to the rigorous upscaling of the infinite adsorption rate reactive flow under dominant Peclet number through a pore

In this article, we undertake a rigorous derivation of the upscaled model for reactive flow through a narrow and long two-dimensional pore. The transported and diffused solute particles undergo the infinite adsorption rate reactions at the lateral tube boundary. At the inlet boundary we suppose Danckwerts’ boundary conditions. The transport and reaction parameters are such that we have dominant Peclet number. Our analysis uses the anisotropic singular perturbation technique, the small characteristic parameter ε being the ratio between the thickness and the longitudinal observation length. Our goal is to obtain error estimates for the approximation of the physical solution by the upscaled one. They are presented in the energy norm. They give the approximation error as a power of ε and guarantee the validity of the upscaled model. We use the Laplace transform in time to get better estimates than in our previous article [Mikelić, Rosier, Rigorous upscaling of the infinite adsorption rate reactive flow under dominant Peclet number through a pore, Ann Univ Ferrara Sez. VII Sci. Mat. 2007 53, 333–359] and to undertake the study of important Danckwerts' boundary conditions.

Rigorous upscaling of the infinite adsorption rate reactive flow under dominant Peclet number through a pore

In this paper we present a rigorous derivation of the effective model for enhanced diffusion through a narrow and long 2D pore. The analysis uses the anisotropic singular perturbation technique. Starting point is a local pore scale model describing the transport by convection and diffusion of a reactive solute. The solute particles undergo an adsorption process at the lateral tube boundary, with high adsorption rate. The transport and reaction parameters are such that we have large, dominant Peclet number with respect to the ratio of characteristic transversal and longitudinal lengths (the small parameter \(\varepsilon\)). We give a formal derivation of the model using the anisotropic multiscale expansion with respect to \(\varepsilon\) . Error estimates for the approximation of the physical solution, by the upscaled one, are presented in the energy norm as well as in L∞ and L1 norms with respect to the space variable. They give the approximation error as a power of \(\varepsilon\) and guarantee the validity of the upscaled model through the rigorous mathematical justification of the effective behavior for small \(\varepsilon\).

Heat equation in a perforated composite plate: influence of a coating

This paper deals with the homogenization of the heat equation in a periodically perforated composite plate. After giving a dimension analysis of the original mathematical model, we are led to an anisotropic singular perturbation boundary-value problem which involves non-standard boundary conditions. There are several dimensionless physical variables like the Biot numbers B p (p=−,+,c − and c +) , the period ε of the distribution of the holes, the respective relative thickness e, δ of the plate and the coating, and ratio λ of the heat conductivities. We only study situations in which the parameters B −, B + and e are of the order of unity and ε≺≺1, δ≺≺1, λ≺≺1.

On a Γ-limit of a family of anisotropic singular perturbations

We construct an example confirming that the interfacial energy density of the Γ-limit of a family of nonconvex functionals, cannot be computed, in general, by assuming that the local behavior of a sequence of vector-valued minimizers near the interface is unidirectional.