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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 (21 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
An Adaptive Finite Element Method for the Incompressible NavierStokes Equations on Timedependent Domains
, 1995
"... Contents 1 Introduction and Notations 1 2 Moving Boundary Problems 9 2.1 Flow in a Channel with a Moving Indentation : : : : : : : : : 9 2.2 Flow in a Water Pump : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Time Discretization : : : : : : : : : : : : : : : : : : : : : : : : 21 2.3.1 Investiga ..."
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Cited by 30 (5 self)
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Contents 1 Introduction and Notations 1 2 Moving Boundary Problems 9 2.1 Flow in a Channel with a Moving Indentation : : : : : : : : : 9 2.2 Flow in a Water Pump : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Time Discretization : : : : : : : : : : : : : : : : : : : : : : : : 21 2.3.1 Investigation of the continuous problem : : : : : : : : : 21 2.3.2 Semidiscretization : : : : : : : : : : : : : : : : : : : : 25 2.3.3 Full discretization : : : : : : : : : : : : : : : : : : : : : 29 3 Adaptive Finite Elements 31 3.1 Adaptive Algorithm : : : : : : : : : : : : : : : : : : : : : : : : 32 3.2 A residualbased error estimator : : : : : : : : : : : : : : : : : 37 3.3 Multigrid method on locally refined meshes : : : : : : : : : : : 40 4 Error estimators for the Stokes Equations 49 4.1 Discretization of the
Tools for Simulating Nonstationary Incompressible Flow via Discretely DivergenceFree Finite Element Models
, 1994
"... Introduction We consider the "usual" NavierStokes equations, u t \Gamma \Deltau + (u \Delta r)u +rp = f ; r \Delta u = 0 ; in\Omega \Theta (0; T ) ; for given force f and viscosity , with prescribed boundary values on @\Omega and an initial condition at t = 0. The variables u and ..."
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Cited by 21 (13 self)
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Introduction We consider the "usual" NavierStokes equations, u t \Gamma \Deltau + (u \Delta r)u +rp = f ; r \Delta u = 0 ; in\Omega \Theta (0; T ) ; for given force f and viscosity , with prescribed boundary values on @\Omega and an initial condition at t = 0. The variables u and p describe the velocity and the pressure of a viscous incompressible flow in a bounded region\Omega ae R 2 . These fundamental equations are of interest to both more theoretical scientists like mathematicians or physicists, and more applied ones like engineers or industrial users. What are the theoretical aspects needed to develop and to implemen
A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a . . .
, 2002
"... ..."
Instantaneous Control of BackwardFacing Step Flows
, 1997
"... In the present paper suboptimal boundary control strategies for the timedependent, incompressible flow over the backwardfacing step are considered. The objective consists in the reduction of the recirculation bubble of the flow behind the step. Several cost functionals are suggested and a frame fo ..."
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Cited by 16 (2 self)
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In the present paper suboptimal boundary control strategies for the timedependent, incompressible flow over the backwardfacing step are considered. The objective consists in the reduction of the recirculation bubble of the flow behind the step. Several cost functionals are suggested and a frame for the derivation of the optimality systems for a general class of cost functionals is presented. Numerical examples are given. 1999 Elsevier Science B.V. and IMACS. All rights reserved. Keywords: Suboptimal control; Backwardfacing step flows; Recirculation bubble 1. Problem formulation The main objective of the present work is the development of a numerically realizable robust control technique in order to reduce the recirculation bubble behind the step, and thus reducing the reattachment length of a backwardfacingstep flow by controlling the flow at the boundary near the edge of the step. For the following presentation it will be convenient to refer to Fig. 1 which depicts the spatial domain# that constitutes the flow region and the subsets of the boundary that we shall refer to. Let (u 1 ,u 2 ,u 3 ) denote the velocity of the fluid in the directions (x 1 ,x 2 ,x 3 ) and let p denote its pressure. The timedependent NavierStokes equations on the spacetime cylinder Q :=#
Adaptive Finite Element Methods for LowMachNumber Flows with Chemical Reactions
 of 30th Computational Fluid Dynamics, von Karman Institute
, 1999
"... this paper. We use the "generalized minimal residual method" (GMRES) of Saad [51] in order to solve the preconditioned system ..."
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Cited by 16 (7 self)
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this paper. We use the "generalized minimal residual method" (GMRES) of Saad [51] in order to solve the preconditioned system
Second order adaptive boundary conditions for exterior flow problems: nonsymmetric stationary flows in two dimensions
 JOURNAL OF MATHEMATICAL FLUID FECHANICS
, 2006
"... We consider the problem of solving numerically the stationary incompressible NavierStokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite subdomain, which leads to the problem of finding so called “artificial boundary conditions” to repl ..."
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Cited by 15 (9 self)
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We consider the problem of solving numerically the stationary incompressible NavierStokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite subdomain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at in…nity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to nonsymmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a selfconsistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude.
Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations
 Journal of Computational Physics
, 2005
"... We consider the calibration of parameters in physical models described by partial differential equations. This task is formulated as a constrained optimization problem with a cost functional of least squares type using information obtained from measurements. An important issue in the numerical solut ..."
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Cited by 14 (5 self)
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We consider the calibration of parameters in physical models described by partial differential equations. This task is formulated as a constrained optimization problem with a cost functional of least squares type using information obtained from measurements. An important issue in the numerical solution of this type of problem is the control of the errors introduced, first, by discretization of the equations describing the physical model, and second, by measurement errors or other perturbations. Our strategy is as follows: We suppose that the user defines an interest functional I which might depend on both the state variable and the parameters and which represents the goal of the computation. First, we propose an a posteriori error estimator which measures the error with respect to this functional. This error estimator is used in an adaptive algorithm to construct economic meshes by local mesh refinement. The proposed estimator requires the solution of an auxiliary linear equation. Second, we address the question of sensitivity. Applying similar techniques as before, we derive quantities which describe the influence of small changes in the measurements on the value of the interest functional. These numbers, which we call relative condition numbers, give additional information on the problem under consideration. They can be computed by means of the solution of the auxiliary problem determined before. Finally, we demonstrate our approach at hand of a parameter calibration problem for a model flow problem. Key words: Parameter estimation, adaptive mesh refinement, sensitivity analysis
Multigrid methods for stabilized nonconforming finite elements for incompressible flow involving the deformation tensor formulation
 Journal of Numerical Mathematics
"... Abstract — Edgeoriented stabilization methods in the framework of discontinuous Galerkin approaches have been recently proposed by Brenner [3] and particularly by Hansbo and Larson [5] for nonconforming finite element discretizations to satisfy a discrete Korn’s inequality. We develop and analyse ..."
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Abstract — Edgeoriented stabilization methods in the framework of discontinuous Galerkin approaches have been recently proposed by Brenner [3] and particularly by Hansbo and Larson [5] for nonconforming finite element discretizations to satisfy a discrete Korn’s inequality. We develop and analyse corresponding multigrid components in combination with local PressureSchurComplement methods and give numerical examples for incompressible newtonian and nonnewtonian fluids.