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An Assessment of Solvers for Saddle Point Problems Emerging from the Incompressible Navier-Stokes Equations

2017
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Release : 2017
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Book Synopsis An Assessment of Solvers for Saddle Point Problems Emerging from the Incompressible Navier-Stokes Equations by : Naveed Ahmed

Download or read book An Assessment of Solvers for Saddle Point Problems Emerging from the Incompressible Navier-Stokes Equations written by Naveed Ahmed. This book was released on 2017. Available in PDF, EPUB and Kindle. Book excerpt: Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

Theoretical Study of the Incompressible Navier-Stokes Equations by the Least-Squares Method

2018-10-24 Science
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Release : 2018-10-24
Genre : Science
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Book Rating : 729/5 ( reviews)

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Book Synopsis Theoretical Study of the Incompressible Navier-Stokes Equations by the Least-Squares Method by : National Aeronautics and Space Adm Nasa

Download or read book Theoretical Study of the Incompressible Navier-Stokes Equations by the Least-Squares Method written by National Aeronautics and Space Adm Nasa. This book was released on 2018-10-24. Available in PDF, EPUB and Kindle. Book excerpt: Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis. Jiang, Bo-Nan and Loh, Ching Y. and Povinelli, Louis A. Glenn Research Center NCC3-233; RTOP 505-90-5K

Bifurcation Analysis of Fluid Flows

2023-06-30 Science
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Release : 2023-06-30
Genre : Science
Kind : eBook
Book Rating : 521/5 ( reviews)

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Book Synopsis Bifurcation Analysis of Fluid Flows by : Henk A. Dijkstra

Download or read book Bifurcation Analysis of Fluid Flows written by Henk A. Dijkstra. This book was released on 2023-06-30. Available in PDF, EPUB and Kindle. Book excerpt: A better understanding of the mechanisms leading a fluid system to exhibit turbulent behavior is one of the grand challenges of the physical and mathematical sciences. Over the last few decades, numerical bifurcation methods have been extended and applied to a number of flow problems to identify critical conditions for fluid instabilities to occur. This book provides a state-of-the-art account of these numerical methods, with much attention to modern linear systems solvers and generalized eigenvalue solvers. These methods also have a broad applicability in industrial, environmental and astrophysical flows. The book is a must-have reference for anyone working in scientific fields where fluid flow instabilities play a role. Exercises at the end of each chapter and Python code for the bifurcation analysis of canonical fluid flow problems provide practice material to get to grips with the methods and concepts presented in the book.

Finite Element Methods for Incompressible Flow Problems

2016-10-27 Mathematics
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Release : 2016-10-27
Genre : Mathematics
Kind : eBook
Book Rating : 500/5 ( reviews)

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Book Synopsis Finite Element Methods for Incompressible Flow Problems by : Volker John

Download or read book Finite Element Methods for Incompressible Flow Problems written by Volker John. This book was released on 2016-10-27. Available in PDF, EPUB and Kindle. Book excerpt: This book explores finite element methods for incompressible flow problems: Stokes equations, stationary Navier-Stokes equations and time-dependent Navier-Stokes equations. It focuses on numerical analysis, but also discusses the practical use of these methods and includes numerical illustrations. It also provides a comprehensive overview of analytical results for turbulence models. The proofs are presented step by step, allowing readers to more easily understand the analytical techniques.

Finite Elements and Fast Iterative Solvers

2014 Mathematics
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Release : 2014
Genre : Mathematics
Kind : eBook
Book Rating : 790/5 ( reviews)

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Book Synopsis Finite Elements and Fast Iterative Solvers by : Howard C. Elman

Download or read book Finite Elements and Fast Iterative Solvers written by Howard C. Elman. This book was released on 2014. Available in PDF, EPUB and Kindle. Book excerpt: A practical graduate text on Scientific Computing with a focus on numerical solution of partial differential equations and numerical linear algebra. This book, and its associated freely downloadable MATLAB software, is relevant to engineers, applied mathematicians, numerical analysts, and people working in interdisciplinary Scientific Computing.

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