Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. I: reaction-diffusion-type problems

We consider singularly perturbed high-order elliptic two-pointboundary value problems of reaction-diffusion type. It is shownthat, on an equidistant mesh, polynomial schemes cannot achievea high order of convergence that is uniform in the perturbationparameter. Piecewise polynomial Galerkin finite-element methodsare then constructed on a Shishkin mesh. Almost optimal convergenceresults, which are uniform in the perturbation parameter, areobtained in various norms. Numerical results are presented fora fourth-order problem. e-mail address: stynes{at}bureau.ucc.ie.     

A priori meshes for singularly perturbed quasilinear two-point boundary value problems

A quasilinear singularly perturbed boundary value problem withoutturning points is used as a model problem to analyse and comparethe Bakhvalov and Shishkin discretization meshes. The Shishkinmeshes are generalized and improved. Received 26 October 1998. Accepted 3 June 1999.     

Finite-element approximation for singularly perturbed nonlinear elliptic boundary-value problems

Numerical solution of a two-dimensional nonlinear singularly perturbed elliptic partial differential equation ∈ Δu = f(x, u), 0 < x, y < 1, with Dirichlet boundary condition is discussed here. The modified Newton method of third-order convergence is employed to linearize the nonlinear problem in place of the standard Newton method. The finite-element method is used to find the solution of the nonlinear differential equation. Numerical results are provided to demonstrate the usefulness of the method.     

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods appliedto boundary value problems for the positive definite Helmholtz-typeproblem –U + ²U = 0 in a bounded or unbounded domain,with the parameter real and possibly large. Applications arisein the implementation of space–time boundary integralmethods for the heat equation, where is proportional to 1/(t),and t is the time step. The corresponding layer potentials arisingfrom this problem depend nonlinearly on the parameter and havekernels which become highly peaked as , causing standard discretizationschemes to fail. We propose a new collocation method with arobust convergence rate as . Numerical experiments on a modelproblem verify the theoretical results.     

A collocation method for singularly perturbed two-point boundary value problems with splines in tension

An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problemay+by+cy=f(·,y),y(0)=y ₀,y(1)=y ₁. Sharp error bounds for the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems (|a|=1), the representation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory.Supported by grant 1-01-254 of the Ministry of Science and Technology, Croatia.     

THE SINGULARLY PERTURBED NONLOCAL BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER NONLINEAR ELLIPTIC EQUATIONS

In this paper,a class of singular perturbation of noidocal boundary value problems forelliptic partial differentia[ equations of higher order is considered by using the differential in-equalities. The uniformly valid asymptotic expansion of solution is obtained.     

The singularly perturbed boundary value problems for semilinear elliptic equation of higher order

The singularly perturbed boundary value problems for the semilinear elliptic equation of higher order are considered. Under suitable conditions and by using the fixed point theorem the existence, uniqueness and asymptotic behavior of solution for the boundary value problems are studied.     

一类半线性椭圆型方程奇摄动Robin边值问题

The singularly perturbed Robin boundary value problems for the semilinear elliptic equation are considered. Under suitable conditions and by using the fixed point theorem the existence ,uniqueness and asymptotic behavior of solution for the boundary value problems are studied.     

奇摄动非线性边值问题

The singularly perturbed nonlinear boundary value problems are considered. Using the stretched variable and the method of boundary layer correction,the formal asymptotic expansion of solution is obtained. And then the uniform validity of solution is proved by using the differential inequalities.     

Parallel two-level Schwarz methods for singularly perturbed elliptic problems

Schwarz domain decomposition methods are developed for the numerical solution of singularly perturbed elliptic problems. Three variants of a two-level Schwarz method with interface subproblems are investigated both theoretically and from the point of view of their computer realization on a distributed memory multiprocessor computer. Numerical experiments illustrate their parallel performance as well as their behavior with respect to the critical parameters such as the perturbation parameter, the size of the interface subdomains and the number of parallel processors. Application of one of the methods to a model problem with an interior layer of complex geometry is also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation ?²Δu = F(u, x, y, ?), where F(u, x, y, ?) is a nonlinear function of u. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.     

Global convergence method for singularly perturbed boundary value problems

We give uniformly convergent splines difference scheme for singularly perturbed boundary value problems(1)$-\epsilon u^{″}+p(x)u^{\prime }+q(x)u=f(x),u(a)={\alpha }_0,u(b)={\alpha }_1,$by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration. We prove the $\mathrm{O}(\min (h^2,{\epsilon }^2))$ order of uniform convergence with respect to small parameter $\epsilon$ at nodes on uniform mesh and $\mathrm{O}(\min (h,\epsilon ))$ order of uniform global convergence with respect to the approximate solution given by $S(x)={\sum }_{i=1}^N{S}_{{\Delta }_i}(x)H(x_i-x)$ where H is the Heaviside function, which is the approximation for the closed form of the exact solution.     

Parameter-uniform numerical methods for singularly perturbed mixed boundary value problems using grid equidistribution

In this paper, we present the analysis of an upwind scheme for obtaining the solution of a convection-diffusion two-point boundary value problem with Robin boundary conditions. The solution is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme converges uniformly with respect to the perturbation parameter. Numerical results are presented that demonstrate the sharpness of the theoretical estimates.     

Galerkin-wavelet methods for two-point boundary value problems

Summary Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.This work was supported by National Science Foundation     

Third-order variable-mesh cubic spline methods for nonlinear two-point singularly perturbed boundary-value problems

A family of third-order variable-mesh methods for singularly perturbed two-point boundary-value problems of the form y=f(x,y,y),y(a)=A, y(b)=B is derived. The convergence analysis is given, and the method is shown to have third-order convergence properties. Several test examples are solved to demonstrate the efficiency of the method.     

Estimate of boundary layer thickness for linear,singularly perturbed two-point boundary-value problems

In this note, the thickness of the boundary layers for linear, homogeneous, singularly perturbed first-order ordinary differential systems is considered. Inequalities concerning this thickness are derived. It is also remarked that the present analysis can be carried over to the nonhomogeneous case.The authors wish to express their sincere thanks to Dr. S. M. Roberts for his comments and valuable suggestions.     

一类具有无穷边界值的二次奇摄动边值问题

研究一类具有无穷边界值的二次奇摄动Robin边值问题解的存在性与解的渐进行为,重点关注边界值的奇异程度对解的边界层行为的影响;同时将所得的结果与Chang及Howes的结果(带正常边界值)进行比较.研究表明:(1)当边界值大小为O(1/)时,得到的边界层大小为O( ln ),这比Chang及Howes带正常边界值的情形提高了O(ln )量级;(2)增大边界值的奇性至O(1/ r),这里r >1,边界层大小的量级不变,依然为O( ln );(3)若要使得边界层大小为O(1),则边界值的大小需为O(e?1/).最后给出一个算例验证得到的结果.