Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data

Summary Error estimates for the semidiserete Galerkin method for abstract semilinear evolution equations with non-smooth initial data are given. In concrete cases almost optimal order of convergence for linear finite elements results.To Professor Dr. J.A. Nitsche on the occasion of his sixtieth birthday     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

Error bounds and estimates for eigenvalues of integral equations. II

Summary In a previous paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of non-symmetric integral equations. In this note an alternative analysis is presented leading to equivalent dominant error terms with error bounds which are quicker to calculate than those derived previously.     

Error estimates for interpolatory quadrature formulae

Summary In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Pólya type formulae.Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainderR[f] of a formula of orderm can be represented asc·f ^(m)(). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases.Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed orderm the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.     

On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations

Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.Supported, in part, by the U.S. Air Force Office of Scientific Research under Grant No. AF-AFOSR-80-0083Supported, in part, by the same agency under Grant No. AF-AFOSR-80-0176-A. Both authors were also partially supported by NASA Contract No. NAS1-15810 while they were in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665, USA     

Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

Summary A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal orderL ^p error estimates are derived for 2p. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.     

Error estimates for the finite element solution of variational inequalities

Summary We study the mixed finite element approximation of variational inequalities, taking as model problems the so called obstacle problem and unilateral problem. Optimal error bounds are obtained in both cases.Supported in part by National Science Foundation grant MCS 75-09457, and by Office of Naval Research grant N00014-76-C-0369     

Error estimates for spline interpolants on equidistant grids

Summary For oddm, the error of them-th-degree spline interpolant of power growth on an equidistant grid is estimated. The method is based on a decomposition formula for the spline function, which locally can be represented as an interpolation polynomial of degreem which is corrected by an (m+1)-st.-order difference term.Dedicated to Prof. Dr. Karl Zeller on the occasion of his 60th birthday     

An analysis of a mixed finite element method for the Navier-Stokes equations

Summary A simple mixed finite element method is developed to solve the steady state, incompressible Navier-Stokes equations in a neighborhood of an isolated—but not necessarily unique—solution. Convergence is established under very mild restrictions on the triangulation, and, when the solution is sufficiently smooth, optimal error bounds are obtained.     

The convergence of two numerical schemes for the Navier-Stokes equations

Summary This paper deals with some convergence/stability results concerning two numerical methods for solving the incompressible nonstationary Navier-Stokes equations. The algorithms are of a particular kind in what regards time discretization (more precisely, of the Peaceman-Rachford and the Strang type resp.), and have been obtained by modifying slightly the numerical treatment of the nonlinear terms in other schemes due to Glowinski et al. (1980). We first describe the full discretization of the homogeneous Dirichlet problem using a (general) external approximation of the spatial functional spaces involved (a particular and simple choice of such an approximation is the standardP ₂-Lagrange finite element for the velocity field when the fluid is bidimensional). Then we establish and prove convergence and stability and make some comments on the numerical treatment of other (generally nonhomogeneous) boundary conditions. The theoretical results show that the schemes are (at least) conditionally stable and convergent, which justifies the success of Glowinski's methods.     

A mixed finite element approximation of the Navier-Stokes equations

Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.     

Combined finite element and spectral approximation of the Navier-Stokes equations

Summary We present a method for the numerical approximation of Navier-Stokes equations with one direction of periodicity. In this direction a Fourier pseudospectral method is used, in the two others a standard F.E.M. is applied. We prove optimal rate of convergence where the two parameters of discretization intervene independently.

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Approximation des équations de Navier-Stokes par une méthode éléments finis-spectrale Fourier

Resumé On présente une méthode d'approximation numérique des équations de Navier-Stokes possédant une direction de périodicité. Dans cette direction une méthode pseudospectrale basée sur des développements en série de Fourier est utilisée, dans les deux autres on applique une méthode d'éléments finis standard. On montre que la convergence est optimale et que les deux paramètres de discrétisation peuvent être choisis de façon indépendante.

     

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Summary The Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.     

Error estimates for the combinedh andp versions of the finite element method

Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.     

Alternating methods for sets of linear equations

Summary A generalization of alternating methods for sets of linear equations is described and the number of operations calculated. It is shown that the lowest number of arithmetic operations is achieved in the SSOR algorithm.     

Spectral approximation of the periodic-nonperiodic Navier-Stokes equations

Summary In order to approximate the Navier-Stokes equations with periodic boundary conditions in two directions and a no-slip boundary condition in the third direction by spectral methods, we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure. The compatibility between these two spaces is checked via an infsup condition. We analyze a spectral and a collocation pseudo-spectral method for the Stokes problem and a collocation pseudo-spectral method for the Navier-Stokes equations. We derive error bounds of spectral type, i.e. which behave likeM ^– whereM depends on the number of degrees of freedom of the method and represents the regularity of the data.     

Duality estimates for the numerical solution of integral equations

Summary We formulate and prove Aubin-Nitsche-type duality estimates for the error of general projection methods. Examples of applications include collocation methods and augmented Galerkin methods for boundary integral equations on plane domains with corners and three-dimensional screen and crack problems. For some of these methods, we obtain higher order error estimates in negative norms in cases where previous formulations of the duality arguments were not applicable.     

Superconvergence of piecewise polynomial Galerkin approximations,for Fredholm integral equations of the second kind

Summary Piecewise polynomial Galerkin approximations for Fredholm integral equations of the second kind are shown to posses superconvergence properties in some circumstances.     

Discrete approximations to spherically symmetric distributions

Summary We consider the approximation of spherically symmetric distributions in ^d by linear combinations of Heaviside step functions or Dirac delta functions. The approximations are required to faithfully reproduce as many moments as possible. We discuss stable methods of computing such approximations, taking advantage of the close connection with Gauss-Christoffel quadrature. Numerical results are presented for the distributions of Maxwell, Bose-Einstein, and Fermi-Dirac.Dedicated to Fritz Bauer on the occasion of his 60th birthdayWork supported in part by the National Science Foundation under Grant MCS-7927158A1     

Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition

Summary We consider a mixed finite element approximation of the stationary, incompressible Navier-Stokes equations with slip boundary condition, which plays an important rôle in the simulation of flows with free surfaces and incompressible viscous flows at high angles of attack and high Reynold's numbers. The central point is a saddle-point formulation of the boundary conditions which avoids the well-known Babuka paradox when approximating smooth domains by polyhedrons. We prove that for the new formulation one can use any stable mixed finite element for the Navier-Stokes equations with no-slip boundary condition provided suitable bubble functions on the boundary are added to the velocity space. We obtain optimal error estimates under minimal regularity assumptions for the solution of the continous problem. The techniques apply as well to the more general Navier boundary condition.