Boundary concentration phenomena for a singularly perturbed elliptic problem

We exhibit new concentration phenomena for the equation ? ε² Δu + u = u^p in a smooth bounded domain Ω ? ?² and with Neumann boundary conditions. The exponent p is greater than or equal to 2 and the parameter ε is converging to 0. For a suitable sequence ε_n → 0 we prove the existence of positive solutions u_n concentrating at the whole boundary of Ω or at some component. © 2002 Wiley Periodicals, Inc.     

一类奇摄动椭圆方程在摄动区域上的渐近解

研究了一类具有摄动边界的非线性椭圆方程摄动问题.经过极坐标变换,在适当的条件下,通过构建近似解以及校正项,利用上下解方法和微分不等式理论得到了解的渐近性态,并通过实际例子进行了验证.     

Asymptotics of the solution to a singularly perturbed elliptic problem with a three-zone boundary layer

For a singularly perturbed elliptic boundary value problem, an asymptotic expansion of the boundary-layer solution is constructed and justified in the case when the boundary layer consists of three zones with different behavior of the solution, which is caused by the multiplicity of the root of the degenerate equation.     

An elliptic singularly perturbed problem with two parameters II: Robust finite element solution

In this paper we consider a singularly perturbed elliptic model problem with two small parameters posed on the unit square. The problem is solved numerically by the finite element method using piecewise linear or bilinear elements on a layer-adapted Shishkin mesh. We prove that method with bilinear elements is uniformly convergent in an energy norm. Numerical results confirm our theoretical analysis.     

A new type of concentration solutions for a singularly perturbed elliptic problem

We prove the existence of positive solutions concentrating on some higher dimensional manifolds near the boundary of the domain for a nonlinear singularly perturbed elliptic problem.

     

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.     

Numerical solution of a mixed singularly perturbed parabolic-elliptic problem

A one-dimensional singularly perturbed problem of mixed type is considered. The domain under consideration is partitioned into two subdomains. In the first subdomain a parabolic reaction-diffusion problem is given and in the second one an elliptic convection-diffusion-reaction problem. The solution is decomposed into regular and singular components. The problem is discretized using an inverse-monotone finite volume method on condensed Shishkin meshes. We establish an almost second-order global pointwise convergence in the space variable, that is uniform with respect to the perturbation parameter.     

Positive solution of singularly perturbed Dirichlet problem with singularities

A class of singularly perturbed boundary value problem with singularities is considered. Introducing the stretched variables, the boundary layer corrective terms near x = 0 and x = 1 are constructed. Under suitable conditions, by using the theory of differential inequalities the existence and asymptotic behavior of solution for boundary value problem are proved, uniformly valid asymptotic expansion of solution with boundary layers are obtained,     

Asymptotic solution of the cauchy problem for a singularly perturbed linear system

We construct the asymptotics of the solution of the Cauchy problem for a degenerate singularly perturbed linear system in the case of multiple spectrum of the principal operator. Nezhin Pedagogic University, Nezhin. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1126–1128, August, 1999,     

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED PROBLEM FOR A NONLINEAR SINGULAR EQUATION

The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.     

Concentration on Hopf-fibres for singularly perturbed elliptic equations

Singularly perturbed elliptic equations with superlinear nonlinearities of polynomial type are considered on an annulus in Rⁿ${\mathbb{R}}^n$, n≥4$n\ge 4$. It is shown that for small parameters there exist solutions which concentrate on manifolds of dimensions one, three and seven, which are given as Hopf-fibres.     

A new non-conforming Petrov-Galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem

A new non-conforming exponentially fitted Petrov-Galerkin finite-elementmethod based on Delaunay triangulation and its Dirichlet tessellationis constructed for the numerical solution of singularly perturbedstationary advectiondiffusion problems with a singular perturbationparameter . The method is analyzed mathematically and its stabilityis shown to be independent of . The error estimate in an -independentdiscrete energy norm for the approximate solution is shown todepend on first-order seminorms of the flux and the zero-orderterm of the equation, the sup norm of the exact solution, thefirst-order seminorm of the coefficient of the advection term,and the approximation error of the inhomogeneous term. Sincethe first two seminorms are not bounded uniformly in , the -uniformconvergence of the method still remains an open question. Noassumption is required that the angles in the triangulationare all acute. Since the system matrix for this method is identicalto that for the exponentially fitted box method, the theoreticalresults also provide an analysis of that box method. The newmethod also contains the central-difference and upwind methodsas two limiting cases. It can be regarded as a weighted finite-differencemethod on a triangular mesh. Numerical results are presentedto show the superior performance of the method in comparisonwith the upwind and central-difference methods for a small increasein the computation cost. Present address: School of Mathematics, The University of NewSouth Wales, Kensington, NSW 2033, Australia.     

Uniform error estimates in the finite element method for a singularly perturbed reaction-diffusion problem

Consider the problem with homogeneous Neumann boundary condition in a bounded smooth domain in . The whole range is treated. The Galerkin finite element method is used on a globally quasi-uniform mesh of size ; the mesh is fixed and independent of .

A precise analysis of how the error at each point depends on and is presented. As an application, first order error estimates in , which are uniform with respect to , are given.

     

Numerical solution of a singularly perturbed problem via exponential splines

It is proved that a spline difference scheme for a singularly perturbed self-adjoint problem, derived by using exponential cubic splines at mid-points, has second order uniform convergence in a small parameter . Numerical experiments are presented to confirm the theoretical predictions.     

奇摄动高阶椭圆型方程解的多重边界层现象

研究了一类奇摄动2m阶椭圆型方程解的多重边层现象．利用比较定理得到解的一致有效的渐近展开式．     

An elliptic singularly perturbed problem with two parameters I: Solution decomposition

In this paper we consider a singularly perturbed elliptic problem with two small parameters posed on the unit square. Its solution may have exponential, parabolic and corner layers. We give a decomposition of the solution into regular and layer components and derive pointwise bounds on the components and their derivatives. The estimates are obtained by the analysis of appropriate problems on unbounded domains.     

Conditions for two-peaked solutions of singularly perturbed elliptic equations

We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain. Received: 20 January 1997 / Revised version: 2 December 1997