On some boundary element methods for the heat equation

Summary The object of this paper is to study some boundary element methods for the heat equation. Two approaches are considered. The first, based on the heat potential, has been studied numerically by previous authors. Here the convergence analysis in one space dimension is presented. In the second approach, the heat equation is first descretized in time and the resulting elliptic problem is put in the boundary formulation. A straight forward implicit method and Crank-Nicolson's method are thus studied. Again convergence in one space dimension is proved.     

A second order splitting method for the Cahn-Hilliard equation

Summary A semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.Research partially supported by NSF Grant DMS-8896141     

A quantitative discrete H2-regularity estimate for the Shortley-Weller scheme in convex domains

Summary We investigate the discreteH ²-regularity properties of the Shortley-Weller discretization of Poisson's equation (with homogeneous Dirichlet boundary condition) in bounded convex domains ². It is shown that the regularity constant is 1 independent of the mesh sizeh if theH ²-seminorm is defined in a way assigning less weight to the (unsymmetric) differences near the boundary.     

A finite element lumped mass scheme for solving eigenvalue problems of circular arches

Summary This study is a continuation of a previous paper [4] in which the numerical results are given by using single precision arithmetic. In this paper, we show the numerical results which experess the sharper convergence properties than those of [4], by using double precision arithmetic.Dedicated to Prof. Masaya Yamaguti on the occasion of his 60th birthday     

A conjugate gradient method and a multigrid algorithm for Morley s finite element approximation of the biharmonic equation

Summary The numerical solution of the linear equations arising from Morley's nonconforming displacement method is studied. A suitable preconditioning for the conjugate gradient method is described. Furthermore, the nonconformity of the discretization necessitates some non-standard ingredients of multigrid algorithms.     

Pointwise error estimates for a streamline diffusion finite element scheme

Summary Pointwise error estimates for a streamline diffusion scheme for solving a model convection-dominated singularly perturbed convection-diffusion problem are given. These estimates improve pointwise error estimates obtained by Johnson et al.[5].     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

A convergence condition for the quadrilateral Wilson element

Summary The paper deals with the convergence properties of the nonconforming quadrilateral wilson element which violates the patch test. The convergence of the element is proved under a certain condition on mesh subdivisions without any modifications of the variational formulation. This result extends the range of applicability of Wilson's element. The necessity of the proposed condition is also discussed.This work was written while the author was visiting the University of Frankfurt, Federal Republic of Germany, on a grant by the Alexander von Humboldt Foundation     

A family of mixed finite elements for the elasticity problem

Summary A new mixed finite element formulation for the equations of linear elasticity is considered. In the formulation the variables approximated are the displacement, the unsymmetric stress tensor and the rotation. The rotation act as a Lagrange multiplier introduced in order to enforce the symmetry of the stress tensor. Based on this formulation a new family of both two-and three-dimensional mixed methods is defined. Optimal error estimates, which are valid uniformly with respect to the Poisson ratio, are derived. Finally, a new postprocessing scheme for improving the displacement is introduced and analyzed.     

A discrete solenoidal finite difference scheme for the numerical approximation of incompressible flows

Summary For a well known class of finite difference schemes for approximating incompressible flows it is shown that the condition of discrete incompressibility can be incorporated into the discrete space. This simplifies the structure of the linear or nonlinear discrete systems and reduces the number of unknowns.     

A modified streamline diffusion method for solving the stationary Navier-Stokes equation

Summary Recently, Hughes et al. [11, 12] proposed new finite element schemes of Petrov-Galerkin type for solving the Stokes problem which do not require the discrete version of the Ladyshenskaya-Babuka-Brezzi-condition (LBB-condition). In this paper we derive a conforming finite element method for solving the stationary Navier-Stokes equations which combines the advantages of arbitrary finite element spaces for velocity/pressure with the favourable properties of the streamline diffusion method in the case of moderate and high Reynolds number.     

A multi-grid algorithm for mixed problems with penalty

Summary In this paper we describe a multi-grid algorithm for the finite element approximation of mixed problems with penalty by the MINI-element. It is proved that the convergence rate of the algorithm is bounded away from 1 independently of the meshsize and of the penalty parameter. For convenience, we only discuss Jacobi relaxation as smoothing operator in detail.The paper was written during the author's stay at the Ruhr-Universität Bochum and revised by D. Braess after the author's return to China     

A constructive method for deriving finite elements of nodal type

Summary In this paper, we propose an algorithm to derive nodal methods corresponding to various two and three-dimensional nonconforming and mixed finite elements. We show that this algorithm can be used to obtain several classical schemes as well as some more recently developed schemes, and that it leads to a simple proof of unisolvence for these methods. Finally we use our method to obtain a three dimensional nodal scheme of BDM type.     

A relaxation procedure for domain decomposition methods using finite elements

Summary We present the convergence analysis of a new domain decomposition technique for finite element approximations. This technique was introduced in [11] and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent. No global preconditioner is needed in the practice, but simply single-domain finite element solvers are required. An optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated. Applications are given to both elliptic equations and incompressible Stokes equations.     

A particle method for first-order symmetric systems

Summary We present and study a conservative particle method of approximation of linear hyperbolic and parabolic systems. This method is based on an extensive use of cut-off functions. We prove its convergence inL ² at the order as soon as the cut-off function belongs toW ^m+1.1.Dedicated to Professor Joachim Nitsche on the occasion of his 60th birthday     

A family of mixed finite elements for linear elasticity

Summary A family of finite elements for use in mixed formulations of linear elasticity is developed. The stresses are not required to be symmetric, but only to satisfy a weaker condition based upon Lagrange multipliers. This is based on the same formulation used in the PEERS finite element spaces. Elements for both two and three dimensional problems are given. Error analysis on these elements is done, and some superconvergence results are proved.     

A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity

Summary In this paper we describe and analyse a numerical method that detects singular minimizers and avoids the Lavrentiev phenomenon for three dimensional problems in nonlinear elasticity. This method extends to three dimensions the corresponding one dimensional method of Ball and Knowles.     

A finite element method for the nonlinear Tricomi problem

Summary We consider a finite element procedure for numerical solution of the nonlinear problem:L[u]=yu _xx +u _yy +r(x,y)u=f(x, y, u) in a simply connected regionG in thex-y plane. The boundary ofG consists of ₀, ₁, and ₂ and we impose the boundary condition . ₀ is assumed to be a piecewises smooth curve lying in the half-planey>0 with endpointsA(–1, 0) andB(0, 0). ₁ and ₂ are characteristics of the operatorL issued fromA andB which intersect at the pointC(–1/2,y _c). An error analysis of the method is also given.     

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.     

A family of higher order mixed finite element methods for plane elasticity

Summary The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL ²(), and optimal order negative norm estimates are obtained inH ^s () for a range ofs depending on the index of the finite element space. An optimal order estimate inL () for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.This work was performed while Professor Arnold was a NATO Postdoctoral Fellow