A finite element scheme for domains with corners

This paper is concerned with the rate of convergence of the finite element method on polygonal domains in weighted Sobolev spaces. It is shown that the use of different spaces of trial and test functions will restrict the usual low rate of convergence to a neighborhood of each vertex of the polygonal domain.L ₂-convergence and lower bounds on the error are also studied.This research was supported in part by the Atomic Energy Commission under contract no. AEC AT-(40-1)-3443/4.This research was supported in part by the U.S. Naval Academy Research Council.     

Solving the biharmonic Dirichlet problem on domains with corners

The biharmonic Dirichlet boundary value problem on a bounded domain is the focus of the present paper. By Riesz' representation theorem the existence and uniqueness of a weak solution is quite direct. The problem that we are interested in appears when one is looking for constructive approximations of a solution. Numerical methods using for example finite elements, prefer systems of second equations to fourth order problems. Ciarlet and Raviart in 7 and Monk in 21 consider approaches through second order problems assuming that the domain is smooth. We will discuss what happens when the domain has corners. Moreover, we will suggest a setting, which is in some sense between Ciarlet‐Raviart and Monk, that inherits the benefits of both settings and that will give the weak solution through a system type approach.     

A mixed finite element method for a nonlinear Dirichlet problem

We study a mixed finite element approximation of a nonlinearDirichlet problem in both two and three dimensions. This studyis a first step towards the treatment of Ladyzhenskaya flowsor quasi-Newtonian flows obeying the power law by mixed finiteelement methods. We give existence and uniqueness results forthe continuous problem and its approximation and we prove anerror bound.     

用单层位势求解尖角区域上的Dirichlet外问题

采用Kress变换以及处理第一类奇异核的积分方法,运用Nystrom方法利用单层位势求解尖角区域上的Dirichlet外问题.给出具体的算法和数值例子,通过数值例子可以看出用单层位势求解尖角区域上的Dirichlet外问题与用单双层结合求解所得的结果基本上一致,说明这种方法是有效的和可行的.     

Improved difference schemes for the Dirichlet problem of Poisson's equation in the neighbourhood of corners

Summary The standard 5-point difference scheme for the model problem u=f on a special polygonal domain with given boundary values is modified at a few points in the neighbourhood of the corners in such a way that the order of convergence at interior points is the same as in the case of a smooth boundary. As a side result improved error bounds for the usual method in the neighbourhood of corners are given.     

Estimates of AGMON-MIRANDA type for solutions of the DIRICHLET problem in plane domains with corners

It is proved that any edge of a 4-connected non-planar graph G of order at least 6 lies in a subdivision of K_3,3 in G. For any 3-connected non-planar graph G of order at least 6 we show that G contains at most four edges which belong to no subdivisions of K_3,3 in G.     

Some remarks on the Dirichlet problem in plane exterior domains

Abstract  In this paper we deal with the Dirichlet problem for the Laplace equation in a plane exterior domain Ω with a Lipschitz boundary. We prove that, if the boundary datum a is square summable, then the problem admits a solution which tends to a in the sense of nontangential convergence, is unique in a suitable function class and vanishes at infinity as r^–k if and only if a satisfies k compatibility conditions, which we are able to explicit when Ω is the exterior of an ellipse. Keywords: Dirichlet problem, Asymptotic behavior, Potential theory Mathematics Subject Classification (2000): 31A05, 31A10     

A new coupling of mixed finite element and boundary element methods for an exterior Helmholtz problem in the plane

This paper deals with the scattering of time harmonic electromagnetic waves by an infinitely long cylinder containing a non-homogeneous conducting medium. More precisely, we study the transverse magnetic field that solves an interface problem holding between the cross section of the cylinder and the exterior two-dimensional free space. We apply a dual-mixed variational formulation in the obstacle coupled with a boundary integral equation method in the unbounded homogeneous space. A Fredholm alternative is utilized to prove that this continuous formulation is well posed. We define the corresponding discrete scheme by using the lowest order rotated Raviart-Thomas finite elements for the magnetic field and spectral elements for the boundary unknown. Then, we show that the resulting Galerkin scheme is uniquely solvable and convergent, and prove optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments. This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics, by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program, and by the Ministerio de Educación y Ciencia of Spain, through the project No. MTM2004-05417.     

On splitting schemes in the mixed finite element method

Within the research into some geothermal modes, a 3D heat transfer process was described by a first-order system of differential equations (in terms of “temperature-heat-flow”). This system was solved by an explicit scheme for the mixed finite element spatial approximations based on the Raviart-Thomas degrees of freedom. In this paper, several algorithms based on the splitting technique for the vector heat-flow equation are proposed. Some comparison results of accuracy of the algorithms proposed are presented.     

The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Dirichlet to Neumann map for domains with corners and approximate boundary conditions

We present in this paper the Dirichlet to Neumann operator for the wave equation on a straight wedge in R²${\mathbb{R}}^2$, using Fourier integral operators. As a consequence, we recover the classical approximate boundary conditions of orders 1 and 2.     

On the mixed boundary-value problem for harmonic functions in plane domains

This paper considers the problem of the determination of a harmonic function in a simply connected plane domain when the values of the function are known on some arcs of the boundary and the values of the normal derivative are known on the remaining boundary. We first present the solution in theoretical form and then show how to compute the solution with the aid of rapidly convergent series. The coefficients of these series are Fourier coefficients of certain functions and can be estimated by using the Fast Fourier Transform. The examples considered in the last part of the paper emphasize the advantages of the method presented in this paper for solving the mixed boundary-value problem as compared to other methods used for this purpose.

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Résumé On considère le problème suivant: trouver une fonction harmonique dans un domaine simplement connexeDR ² dont la frontièreS est assez régulière, en connaissant les valeurs de la fonction sur une partieS ₁ de la frontière et les valeurs de la dérivée normale sur la complémentaire de366-1. La première partie présente des résultats théoriques liés à ce probléme. La deuxième partie s'appuie sur une représentation de la solution à l'aide de certaines séries de fonctions rapidement convergentes. Les coefficients des séries utilisées sont même les coefficients de Fourier de certaines fonctions, et leur calcul peut être effectué en utilisant la transformation de Fourier rapide (Fast Fourier Transform). Les exemples considérés dans la dernière partie de l'article mettent en évidence les avantages de la méthode présentée par rapport à d'autres méthodes employées pour la résolution du problème mixte.

     

On finite element schemes of the dirichlet problem

In this paper, we consider finite element schemes applied to the Dirichlet problem for the system of nonlinear elliptic equations, based on piecewise linear polynomials, and present iterative methods for solving algebraic nonlinear equations, which construct monotone sequences. Furthermore, we derive error estimates which imply uniform convergence. Our results are based on the discrete maximum principle. Finally, some typical numerical examples are given to demonstrate the usefulness of convergence results.     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

L 2 estimates for the finite element method for the Dirichlet problem with singular data

Summary In this paper we consider the approximation by the finite element method of second order elliptic problems on convex domains and homogeneous Dirichlet condition on the boundary. In these problems the data are Borel measures. Using a quasiuniform mesh of finite elements and polynomials of degree 1, we prove that in two dimensions the convergence is of orderh inL ² and in three dimensions of orderh ^1/2.     

3-D mixed finite element schemes for charge transport equations

We use the Raviart-Thomas-Nédéléc space to discretize the current continuity equations of the drift-diffusion semiconductor models with Mixed Finite Element methods in ³. An asymptotic analysis of the behaviour od the scheme when the potential is very large is given.