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An overview of projection methods for incompressible flows
 Comput. Methods Appl. Mech. Engrg
"... Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1no ..."
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Cited by 203 (20 self)
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Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1norm for the velocity and of the L 2norm for the pressure. We also show that the class of fractionalstep methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75–111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414–443] can be interpreted as the rotational form of our velocitycorrection methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L 2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods. Key words. Navier–Stokes equations, projection methods, fractionalstep methods, incompressibility, finite elements, spectral approximations
Spectral Problems for the Lamé System with Spectral Parameter in Boundary Conditions on Smooth or Nonsmooth Boundary
, 1999
"... . The paper is devoted to four spectral problems for the Lame system of linear elasticity in domains of R 3 with compact connected boundary S. The frequency is xed in the upper closed halfplane; the spectral parameter enters into the boundary or transmission conditions on S. Two cases are inve ..."
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Cited by 5 (2 self)
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. The paper is devoted to four spectral problems for the Lame system of linear elasticity in domains of R 3 with compact connected boundary S. The frequency is xed in the upper closed halfplane; the spectral parameter enters into the boundary or transmission conditions on S. Two cases are investigated: (1) S is C 1 ; (2) S is Lipschitz. INTRODUCTION In this paper we consider four spectral problems for the Lame system of linear elasticity, see (1.3). The system contains the frequency parameter !, which is a xed complex number with Re! > 0. The statements of Problems I{IV are given in Subsection 1.1. The spectral parameter enters into the boundary conditions (in Problems I, II) or transmission conditions (in Problems III, IV) on a closed connected surface S, which divides its complement into a bounded domain G + and an unbounded domain G . This surface is assumed to be innitely smooth in Section 1 and Lipschitz in Section 2. Our aim is to study the spectral properties ...
Propagation of cracks in elastic media
 Arch. Rational Mech. Anal
, 1996
"... In this paper we consider a quasistationary model of crack propagation in a twodimensional elastic medium occupying a bounded domain. The model, developed in [3], is based on earlier work [1], [4], [9], [10], [11]. The motion of the tip X(t) of the crack (t) at time t is given by X(t) _ = (j _ X ..."
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Cited by 2 (1 self)
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In this paper we consider a quasistationary model of crack propagation in a twodimensional elastic medium occupying a bounded domain. The model, developed in [3], is based on earlier work [1], [4], [9], [10], [11]. The motion of the tip X(t) of the crack (t) at time t is given by X(t) _ = (j _ X(t)j)J(X(t)) (1.1) where (s) is an explicit (and rather simple) function of s and J(X(t)) is de ned in terms of the, so called, Jintegral. The stress function '(x; t) satis es: ' 2 H 2 ( ) for t>0; (1.2) 2 ' =0 in n (t); (1.3)
PERIODIC SOLUTIONS OF THE EQUATIONAv: v(1I v2) IN • AND •
"... We study in this paper the existence of periodic functions v ß I • • C which satisfy the equation (1)v " v(1 Ivl). As observed in [2], the functions ..."
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We study in this paper the existence of periodic functions v ß I • • C which satisfy the equation (1)v " v(1 Ivl). As observed in [2], the functions