Elliptic Singular Perturbation Problems Research Articles

A robust nonconforming $H^2$-element

Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming H 2 -element which is H 1 -conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.

The $p$ and $hp$ versions of the finite element method for problems with boundary layers

We study the uniform approximation of boundary layer functions exp(-x/d) for x ∈ (0,1), d ∈ (0,1], by the p and hp versions of the finite element method. For the p version (with fixed mesh), we prove super-exponential convergence in the range p + 1/2 > e/(2d). We also establish, for this version, an overall convergence rate of O(p -1 √ln p) in the energy norm error which is uniform in d, and show that this rate is sharp (up to the √ln p term) when robust estimates uniform in d ∈ (0,1] are considered. For the p version with variable mesh (i.e., the hp version), we show that exponential convergence, uniform in d ∈ (0,1], is achieved by taking the first element at the boundary layer to be of size O(pd). Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when d is as small as, e.g., 10 -8 . They also illustrate the superiority of the hp approach over other methods, including a low-order h version with optimal exponential mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

A hyperbolic approach to elliptic and parabolic singular perturbation problems

A hyperbolic approach to elliptic and parabolic singular perturbation problems

Remarks on elliptic singular perturbation problems

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton—Jacobi—Bellman equations with a small parameter.

The exponential leveling in elliptic singular perturbation problems with complicated attractors

The exponential leveling in elliptic singular perturbation problems with complicated attractors

Semilinear elliptic singular perturbation problems with nonuniform interior behavior

Pour un probleme elliptique de valeurs aux limites de la forme Lu+f(χ,u)=0, dans Ω, Bu=h sur ∂Ω, Matano (1979) avait montre l'existence d'une solution intermediaire entre deux solutions stables quelconques. Ici on etudie ce type de solution pour le probleme de Dirichlet. e 2 Δu+f(χ,u)=0 dans Ω, u=0 sur ∂Ω, ou e est un petit parametre, Ω est un domaine borne dans R n avec une frontiere du type C 2+ α pour un α∈(0,1)

High‐order methods for differential equations with large first‐derivative terms

AbstractA method is developed to solve elliptic singular perturbation problems. Examples are presented in one and two dimensions for both linear and non‐linear problems. In particular, examples are presented for fluid flow problems with boundary layers. In the one‐dimensional case an approximating equation is developed using just three points. The method first presented is a fourth‐order approximation but is extended to become a higher‐order method. Results are included for the fourth‐, sixth‐, eighth‐ and tenth‐order methods.The results are first compared with results found by Segal in an article about elliptic singular perturbation problems. The elliptic singular perturbation problems are compared with a method by Il'in and also with central and backward difference schemes from Segal's article. There was only one case where the results in Segal's paper were as accurate as the results presented in this paper. However, in this case the method used by Segal did not give accurate values for a second problem presented. The results are also compared with results given by Spalding and by Christie.The method of this paper was also tested on the solution of some non‐linear diffusion equations with concentration‐dependent diffusion coefficients. The results were superior to results presented by Lee and by Schultz. Finally, the method is extended to several two‐dimensional problems.The method developed in this paper is accurate, easy to use and can be generalized to other problems.

Exponential descent of solutions of elliptic singular perturbation problems

Soit Ω un ouvert de R m a frontiere lisse ∂Ω. On considere les solutions du probleme de Dirichlet eΔu+Σ 1=1 m b i u xi +(cu=0) dans Ω u/ ∂Ω =φ(x). On etudie le comportement asymptotique des solutions ue(x) quand e→o

Aspects of Numerical Methods for Elliptic Singular Perturbation Problems

Upwind difference, defect correction and central difference schemes for the solution of the convection-diffusion equation with small viscosity coefficient are compared. It is shown that central difference schemes and hence also standard Galerkin finite element methods are preferable above upwind and defect correction schemes, when Gaussian elimination is used for the solution of the resulting system of equations.When iterative solution methods are employed good results can be achieved by a defect-correction method, whereas upwind difference schemes are generally inaccurate.

On Elliptic Singular Perturbation Problems with Turning Points

The boundary value problem for the elliptic equation $\varepsilon \Delta u + \sum {b_i u_{x_i} } = 0$ is considered in the case that the characteristic curves of the reduced equation enter the domain and have a singular point inside (turning point). Assume that there exists a potential function $\psi (x)$ such that $b_i = \psi _{x_i} (i = 1,2, \cdots n)$. It is proved that if $\varepsilon \to 0$ then the solutions $u_\varepsilon (x)$ converge to a constant, a formula for which was derived by Matkowsky and Schuss using formal asymptotic expansion.

On a class of elliptic singular perturbations with applications in population genetics

With the maximum principle for differential equations asymptotic estimates are made for a class of linear elliptic singular perturbation problems with resonant turning point behaviour in some of the independent variables. The method is applied to the stationary solution of the Kolmogorov backward equation in population genetics.

Some aspects of non-uniform convergence in an elliptic singular perturbation problem

In elliptic singular perturbation problems several types of boundary layers occur. A model problem is investigated and it is shown that, by extension of the familiar stretching technique to parameters, uniform results can be obtained.

An elliptic singular perturbation problem with almost characteristic boundaries

An elliptic singular perturbation problem with almost characteristic boundaries

Boundary Layers in Linear Elliptic Singular Perturbation Problems

This paper presents a survey of recent results obtained in the theory of singular perturbations for linear elliptic partial differential equations in two independent variables. The emphasis is on the constructive procedures and the various aspects of the boundary layer phenomenon.

A uniform asymptotic expansion of the solution of a linear elliptic singular perturbation problem

A uniform asymptotic expansion of the solution of a linear elliptic singular perturbation problem