Results 1  10
of
16
A twolevel additive Schwarz preconditioner for nonconforming plate elements
 Numer. Math
, 1994
"... Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree no ..."
Abstract

Cited by 44 (5 self)
 Add to MetaCart
Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap. 1.
Convergence of nonconforming multigrid methods without full elliptic regularity
 Math. Comp
, 1995
"... Abstract. We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the Wcycle algorithm which is independent ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
Abstract. We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the Wcycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable Vcycle algorithm is an optimal preconditioner. 1.
Analysis of a class of nonconforming finite elements for crystalline microstructures
 Math. Comp
, 1996
"... Abstract. An analysis is given for a class of nonconforming Lagrangetype finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure. These elements were first proposed and analyzed i ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
Abstract. An analysis is given for a class of nonconforming Lagrangetype finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure. These elements were first proposed and analyzed in 1992 by Rannacher and Turek for the Stokes equation. Our analysis highlights the features of these elements which make them effective for the computation of microstructure. New results for superconvergence and numerical quadrature are also given. 1.
A robust nonconforming H 2 –element
 Math. Comp
, 2001
"... Abstract. 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 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract. 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 H2element which is H1conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter. 1.
A discontinuous Galerkin method for the plate equation
, 2000
"... We present a discontinuous Galerkin method for the plate problem. The method employs a discontinuous approximation space allowing, non matching grids and different types of approximation spaces. Continuity is enforced weakly through the variational form. Discrete approximations of the normal and ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We present a discontinuous Galerkin method for the plate problem. The method employs a discontinuous approximation space allowing, non matching grids and different types of approximation spaces. Continuity is enforced weakly through the variational form. Discrete approximations of the normal and twisting moments and the transversal force, which satisfy the equilibrium condition on an element level, occur naturally in the method. We show optimal a priori error estimates in various norms and investigate locking phenomena when certain stabilization parameters tend to infinity. Finally, we relate the method to two classical elements; the nonconforming Morley element and the C¹ Argyris element.
Preconditioned Iterative Methods for Scattered Data Interpolation
, 2001
"... ... this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
... this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis functions for the equations corresponding to the interpolation conditions. The effect of the preconditioners are documented by numerical experiments.
Global superconvergence of Adini’s elements coupled with the Trefftz method for singular problems
, 2002
"... This paper reports new results of the Trefftz method (i.e. the boundary approximation method (BAM) by Li [Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, 1998] using particular solutions which is coupled with Adini’s method for singular problems. First, the com ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This paper reports new results of the Trefftz method (i.e. the boundary approximation method (BAM) by Li [Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities, 1998] using particular solutions which is coupled with Adini’s method for singular problems. First, the computational aspects and numerical experiments are provided for the global superconvergence of Adini’s elements for the Poisson equation. The superconvergence O(h 3.5) is obtained over the entire domain for the uniform rectangulation, by means of an a posteriori interpolant of the obtained solutions. Such a superconvergence is a half order higher than the optimal convergence O(h 3) of Adini’s elements. Second, for the Neumann problems, we add the natural boundary constraints ðunÞij gij to the admissible functions of Adini’s elements, the global superconvergence O(h 4) can also be achieved. Third, for singular problems, e.g. the Motz problem, the Adini’s elements are coupled with the Trefftz method also to give the global superconvergence O(h 3.5). This paper reports the numerical results which have verified perfectly the high global superconvergence of Adini’s elements and their coupling with the Trefftz method. For Motz’s problem, the leading coefficient, ~d0 401:1624462 with the relative errors 0:189 £ 1027; is obtained at N 16; e.g. h 1=16: This is the most accurate leading coefficient ever published by the coupled method of the Trefftz method with other different methods [Engng Anal
EQUIVALENCE OF FINITE ELEMENT METHODS FOR PROBLEMS IN ELASTICITY*
"... Abstract. Modifications of the Morley method for the approximation of the biharmonic equation are obtained from various finite element methods applied to the equations of linear isotropic elasticity and the stationary Stokes equations, by elimination procedures analogous to those used in the continu ..."
Abstract
 Add to MetaCart
Abstract. Modifications of the Morley method for the approximation of the biharmonic equation are obtained from various finite element methods applied to the equations of linear isotropic elasticity and the stationary Stokes equations, by elimination procedures analogous to those used in the continuous case. Problems with Korn’s first inequality for nonconforming P1 elements and its implications for the approximation of the elasticity equations are also discussed. Key words, biharmonic, stokes, elasticity, finite element AMS(MOS) subject classifications. 65N30, 73K25 1. Introduction. It
EXTENSION THEOREMS FOR PLATE ELEMENTS WITH APPLICATIONS
"... Abstract. Extension theorems for plate elements are established. Their applications to the analysis of nonoverlapping domain decomposition methods for solving the plate bending problems are presented. Numerical results support our theory. 1. ..."
Abstract
 Add to MetaCart
Abstract. Extension theorems for plate elements are established. Their applications to the analysis of nonoverlapping domain decomposition methods for solving the plate bending problems are presented. Numerical results support our theory. 1.
NONCONFORMING TETRAHEDRAL FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC EQUATIONS
"... Abstract. 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 quasiconformi ..."
Abstract
 Add to MetaCart
Abstract. 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 quasiconforming tetrahedral element. These elements are proved to be convergent for a model biharmonic equation in three dimensions. In particular, the quasiconforming 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. 1.