Uniformly convergent C 0‐nonconforming triangular prism element for fourth‐order elliptic singular perturbation problem

In this article, we introduce a C ⁰‐nonconforming triangular prism element for the fourth‐order elliptic singular perturbation problem in three dimensions by using the bubble functions. The element is proved to be convergent in the energy norm uniformly with respect to the perturbation parameter. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1785–1796, 2014     

Nonconforming tetrahedral finite elements for fourth order elliptic equations

This paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operators in three spatial dimensions. The newly constructed elements include two nonconforming tetrahedral finite elements and one quasi-conforming tetrahedral element. These elements are proved to be convergent for a model biharmonic equation in three dimensions. In particular, the quasi-conforming tetrahedron element is a modified Zienkiewicz element, while the nonmodified Zienkiewicz element (a tetrahedral element of Hermite type) is proved to be divergent on a special grid.

     

Multiple solutions to singular fourth order elliptic equations on compact manifolds

Using the method of Nehari manifolds, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singularities and with critical Sobolev growth.     

The Morley element for fourth order elliptic equations in any dimensions

In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general n-dimensional Morley element consists of all quadratic polynomials defined on each n-simplex with degrees of freedom given by the integral average of the normal derivative on each (n-1)-subsimplex and the integral average of the function value on each (n-2)-subsimplex. Explicit expressions of nodal basis functions are also obtained for this element on general n-simplicial grids. Convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation. The work was supported by the National Natural Science Foundation of China (10571006). This work was supported in part by NSF DMS-0209497 and NSF DMS-0215392 and the Changjiang Professorship through Peking University.     

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems.In conjunction with the optimal meshes,the numerical solutions resulting from the method have optimal convergence order.Numerical experiments are presented to verify our theoretical estimates.     

High accuracy nonconforming finite elements for fourth order problems

The approach of nonconforming finite element method admits users to solve the partial differential equations with lower complexity,but the accuracy is usually low.In this paper,we present a family of highaccuracy nonconforming finite element methods for fourth order problems in arbitrary dimensions.The finite element methods are given in a unified way with respect to the dimension.This is an effort to reveal the balance between the accuracy and the complexity of finite element methods.     

An equation decomposition method for the numerical solution of a fourth‐order elliptic singular perturbation problem

In this article, we propose a tailored finite point method (TFPM) for the numerical solution of a type of fourth‐order singular perturbation problem in two dimensions based on the equation decomposition technique. Our finite point method has been tailored based on the local exponential basis functions. Furthermore, our TFPM satisfies the discrete maximum principle automatically. Our numerical examples show that our method has second order convergence rate in energy norm as $\varepsilon\to0$ . © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

High order generalized upwind schemes and numerical solution of singular perturbation problems

High even order generalizations of the traditional upwind method are introduced to solve second order ODE-BVPs without recasting the problem as a first order system. Both theoretical analysis and numerical comparison with central difference schemes of the same order show that these new methods may avoid typical oscillations and achieve high accuracy. Singular perturbation problems are taken into account to emphasize the main features of the proposed methods. AMS subject classification (2000)  65L10, 65L12, 65L50     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On sensitive elliptic singular perturbation problems and logarithmic oscillations: the case of shells with edges

This work deals with singular perturbation problems depending on small positive parameter ?. The limit problem as ? → 0 has no solution within the classical theory of PDEs, which uses distribution theory. A very particular and less‐known phenomenon appears: large oscillations. These problems exhibit some kind of instability; very small and smooth variations of the data imply large singular perturbations of the solution. That kind of problems appears in elasticity for highly compressible two‐dimensional bodies and thin shells with elliptic middle surface with a part of the boundary free. Here, we consider certain properties of that oscillations and extend the theory to shells with edges. Copyright © 2012 John Wiley & Sons, Ltd.     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Subcritical perturbation of a locally periodic elliptic operator

We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale ? . We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter ? tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is non‐degenerate at this point. Copyright © 2016 John Wiley & Sons, Ltd.     

Convergence of one-leg methods for singular perturbation problems with delays

This paper is concerned with the error behaviour of one-leg methods applied to some classes of one-parameter multiple stiff singularly perturbed problems with delays. We obtain convergence results of A-stable one-leg methods with linear interpolation procedure. Numerical experiments further confirm our theoretical analysis.     

一类四阶奇异边值问题的正解存在的充分必要条件

利用上下解方法和极大值原理给出了一般边界条件下四阶微分方程的奇异迫值问题有C^2[0，1]和C^3[0，1]正解存在的充分必要条件．推广了韦忠礼(1999)的结果。     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

Infinitely many solutions to fourth order superlinear periodic problems

We present a new min-max approach to the search of multiple -periodic solutions to a class of fourth order equations

where is continuous, -periodic in and satisfies a superlinearity assumption when . For every , we prove the existence of a -periodic solution having exactly zeroes in .

     

Numerical integration of a class of singular perturbation problems

In this paper, we discuss an approximate method for the numerical integration of a class of linear, singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This method requires a minimum of problem preparation and can be implemented easily on a computer. We replace the original singular perturbation problem by an approximate first-order differential equation with a small deviating argument. Then, we use the trapezoidal formula to obtain the three-term recurrence relationship. Discrete invariant imbedding algorithm is used to solve a tridiagonal algebraic system. The stability of this algorithm is investigated. The proposed method is iterative on the deviating argument. Several numerical experiments have been included to demonstrate the efficiency of the method.The authors wish to express their sincere thanks to Dr. S. M. Roberts for his comments and valuable suggestions.     

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．