Interior maximum norm estimates for finite element discretizations of the Stokes equations

Interior estimates are proved in the L ^∞ norm for stable finite element discretizations of the Stokes equations on translation invariant meshes. These estimates yield information about the quality of the finite element solution in subdomains a positive distance from the boundary. While they have been established for second-order elliptic problems, these interior, or local, maximum norm estimates for the Stokes equations are new. By applying finite differenciation methods on a translation invariant mesh, we obtain optimal convergence rates in the mesh size h in the maximum norm. These results can be used for analyzing superconvergence in finite element methods for the Stokes equations.     

Analyticity estimates for the Navier–Stokes equations

We study spatial analyticity properties of solutions of the three-dimensional Navier–Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier–Stokes equations with data in Hr, r?1/2, and prove a stability result for the analyticity radius.     

Boundary arcs for integral equations

Behavior of boundary arcs for control systems is investigated when the systems are governed by integral equations of the Volterra type. The main result is in the form of a maximum principle. This result is then used to obtain necessary conditions for a minimum control problem.     

Stokes problem and Muskhelishvili integral equations

Muskhelishvili integral equations are derived for the Stokes problem of steady-state plane viscous flow in an arbitrary bounded simply connected region D. The characteristics of some FORTRAN modules for solving these equations are given and the numerical solution of the Stokes problem for a disk is described.Institute of Mathematics, Ukrainian Academy of Sciences, Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 69–76, 1989.     

Sharp estimates for the maximum over minimum modulus of rational functions

Let 1$">, and be a rational function with numerator, denominator of degree , respectively. In several applications, one needs to know the size of the set such that for ,

In an earlier paper, we showed that

where denotes linear Lebesgue measure. Here we obtain, for each , the sharp version of this inequality in terms of condenser capacity. In particular, we show that as ,

     

A posteriori estimates for approximations of time-dependent Stokes equations

Charalambos Makridakis ^{In this paper, we derive a posteriori error estimates for space-discreteapproximations of the time-dependent Stokes equations. By usingan appropriate Stokes reconstruction operator, we are able towrite an auxiliary error equation, in pointwise form, that satisfiesthe exact divergence-free condition. Thus, standard energy estimatesfrom partial differential equation theory can be applied directly,and yield a posteriori estimates that rely on available correspondingestimates for the stationary Stokes equation. Estimates of optimalorder in L(L2) and L(H1) for the velocity are derived for finite-elementand finite-volume approximations.}     

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.     

Boundary integral method for Stokes flow past a porous body

In this paper we obtain an indirect boundary integral method in order to prove existence and uniqueness of the classical solution to a boundary value problem for the Stokes–Brinkman-coupled system, which describes an unbounded Stokes flow past a porous body in terms of Brinkman's model. Therefore, one assumes that the flow inside the body is governed by the continuity and Brinkman equations. Some asymptotic results in both cases of large and, respectively, of low permeability are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

Singular integral operator,Hardy–Morrey space estimates for multilinear operators and Navier–Stokes equations

After establishing the molecule characterization of the Hardy–Morrey space, we prove the boundedness of the singular integral operator and the Riesz potential. We also obtain the Hardy–Morrey space estimates for multilinear operators satisfying certain vanishing moments. As an application, we study the existence and the uniqueness of the solutions to the Navier–Stokes equations for the initial data in the Hardy–Morrey space ????(p?n) for q as small as possible. Here, the Hardy–Morrey space estimates for multilinear operators are important tools. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A maximum modulus estimate for solutions of the Navier–Stokes system in domains of polyhedral type

The authors prove a maximum modulus estimate for solutions of the stationary Navier–Stokes system in bounded domains of polyhedral type (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value methods for Volterra integral and integro-differential equations

New and effective quadrature rules generated by boundary value methods are introduced. We employ the introduced quadrature rules to construct quadrature methods for the second kind Volterra integral equations and Volterra integro-differential equations. These methods are shown to be effective and possess excellent convergence properties. The nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme. Numerical simulations are presented and confirm the efficiency and accuracy of the methods.     

Boundary decay estimates for solutions of fourth-order elliptic equations

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.     

Boundary integral equations for a body with inhomogeneous inclusions

On the basis of the partially singular differential equations of the stationary problem of heat conduction and the quasi-static problem of thermoelasticity, written taking account of conditions of nonideal thermomechanical contact, we derive boundary integral equations for a body with inhomogeneous inclusions. We propose a method of solving these equations taking account of the order of the principal term of the asymptotics of the solution in neighborhoods of the corners of the contact surfaces. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 37–41.     

Boundary integral equations for screen problems in IR3

Here we present a new solution procedure for Helmholtz and Laplacian Neumann screen or Dirichlet screen problems in IR³ via boundary integral equations of the first kind having as unknown the jump of the field or of its normal derivative, respectively, across the screen S. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problems. Via the Wiener-Hopf method in the halfspace, localization and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behavior near the edge of the screen. We give Galerkin schemes based on our integral equations on S and obtain high convergence rates by using special singular elements besides regular splines as test and trial functions.