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Point Problem (point + problem)

Distribution by Scientific Domains

Engineering	62%
Mathematics and Statistics	37%

Kinds of Point Problem

saddle point problem

Selected Abstracts

Accelerating the convergence of coupled geomechanical-reservoir simulations

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 10 2007
L. Jeannin
Abstract The pressure variations during the production of petroleum reservoir induce stress changes in and around the reservoir. Such changes of the stress state can induce marked deformation of geological structures for stress sensitive reservoirs as chalk or unconsolidated sand reservoirs. The compaction of those reservoirs during depletion affects the pressure field and so the reservoir productivity. Therefore, the evaluation of the geomechanical effects requires to solve in a coupling way the geomechanical problem and the reservoir multiphase fluid flow problem. In this paper, we formulate the coupled geomechanical-reservoir problem as a non-linear fixed point problem and improve the resolution of the coupling problem by comparing in terms of robustness and convergence different algorithms. We study two accelerated algorithms which are much more robust and faster than the conventional staggered algorithm and we conclude that they should be used for the iterative resolution of coupled reservoir-geomechanical problem. Copyright © 2006 John Wiley & Sons, Ltd. [source]

PID adaptive control of incremental and arclength continuation in nonlinear applications

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2009
A. M. P. Valli
Abstract A proportional-integral-derivative (PID) control approach is developed, implemented and investigated numerically in conjunction with continuation techniques for nonlinear problems. The associated algorithm uses PID control to adapt parameter stepsize for branch,following strategies such as those applicable to turning point and bifurcation problems. As representative continuation strategies, incremental Newton, Euler,Newton and pseudo-arclength continuation techniques are considered. Supporting numerical experiments are conducted for finite element simulation of the ,driven cavity' Navier,Stokes benchmark over a range in Reynolds number, the classical Bratu turning point problem over a reaction parameter range, and for coupled fluid flow and heat transfer over a range in Rayleigh number. Computational performance using PID stepsize control in conjunction with inexact Newton,Krylov solution for coupled flow and heat transfer is also examined for a 3D test case. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Two preconditioners for saddle point problems in fluid flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2007
A. C. de Niet
Abstract In this paper two preconditioners for the saddle point problem are analysed: one based on the augmented Lagrangian approach and another involving artificial compressibility. Eigenvalue analysis shows that with these preconditioners small condition numbers can be achieved for the preconditioned saddle point matrix. The preconditioners are compared with commonly used preconditioners from literature for the Stokes and Oseen equation and an ocean flow problem. The numerical results confirm the analysis: the preconditioners are a good alternative to existing ones in fluid flow problems. Copyright © 2006 John Wiley & Sons, Ltd. [source]

A comparative study of efficient iterative solvers for generalized Stokes equations

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2008
Maxim Larin
Abstract We consider a generalized Stokes equation with problem parameters ,,0 (size of the reaction term) and ,>0 (size of the diffusion term). We apply a standard finite element method for discretization. The main topic of the paper is a study of efficient iterative solvers for the resulting discrete saddle point problem. We investigate a coupled multigrid method with Braess,Sarazin and Vanka-type smoothers, a preconditioned MINRES method and an inexact Uzawa method. We present a comparative study of these methods. An important issue is the dependence of the rate of convergence of these methods on the mesh size parameter and on the problem parameters , and ,. We give an overview of the main theoretical convergence results known for these methods. For a three-dimensional problem, discretized by the Hood,Taylor ,,2,,,1 pair, we give results of numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Two preconditioners for saddle point problems in fluid flows

Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier,Stokes equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2002
Volker John
Abstract This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier,Stokes equations within the DFG high-priority research program flow simulation with high-performance computers by Schafer and Turek (Vol. 52, Vieweg: Braunschweig, 1996). The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid method proves to be the best solver. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Two classes of multisecant methods for nonlinear acceleration

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2009
Haw-ren Fang
Abstract Many applications in science and engineering lead to models that require solving large-scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi-Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allow to take into account a variable number of secant equations at each iteration. The first is the Broyden-like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola,Nevanlinna-type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self-consistent field (SCF) iteration, is accelerated by various strategies termed ,mixing'. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A preconditioner for generalized saddle point problems: Application to 3D stationary Navier-Stokes equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2006
C. Calgaro
Abstract In this article we consider the stationary Navier-Stokes system discretized by finite element methods which do not satisfy the inf-sup condition. These discretizations typically take the form of a variational problem with stabilization terms. Such a problem may be transformed by iteration methods into a sequence of linear, Oseen-type variational problems. On the algebraic level, these problems belong to a certain class of linear systems with nonsymmetric system matrices ("generalized saddle point problems"). We show that if the underlying finite element spaces satisfy a generalized inf-sup condition, these problems have a unique solution. Moreover, we introduce a block triangular preconditioner and we show how the eigenvalue bounds of the preconditioned system matrix depend on the coercivity constant and continuity bounds of the bilinear forms arising in the variational problem. Finally we prove that the stabilized P1-P1 finite element method proposed by Rebollo is covered by our theory and we show that the condition number of the preconditioned system matrix is independent of the mesh size. Numerical tests with 3D stationary Navier-Stokes flows confirm our results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006 [source]