Domain decomposition for spectral approximation to stokes equations via divergence-free functions

Pasquarelli, Franco

doi:10.1016/0168-9274(91)90111-c

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…We refer, in particular, to the method of Moser, Moin and Leonard (1983) and its extensions (see Pasquarelli, Quarteroni and Sacchi Landriani (1987) and Pasquarelli (1991)). Once UN is available, the problem is how to recover a pressure field PN with the same optimal accuracy.…”

Section: Approximation By Spectral Methods 319mentioning

confidence: 99%

Numerical Approximation of Partial Differential Equations

Quarteroni

Valli

1994

Springer Series in Computational Mathematics

1,468

1,496

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This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method

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Section: Approximation By Spectral Methods 319mentioning

confidence: 99%

Numerical Approximation of Partial Differential Equations

Quarteroni

Valli

1994

Springer Series in Computational Mathematics

1,468

1,496

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show abstract

“…The main disadvantage of the analytic solenoidal bases is the limitation to simple geometries in the current formulation. However, there are domain decomposition approaches (see [18]) as a possible remedy. Dual "2-basis":ṽ…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…The use of solenoidal functions in fluid mechanics problems appeared earlier in various studies such as [16] in the study of Taylor-Couette flow, and in the study of pipe flow [11,14]. The convergence properties of expansions in divergence-free functions in the case of Stokes equations along with pressure reconstruction algorithm are studied in [19], and a domain decomposition approach can be found in [18]. Another early formulation of the incompressible fluid flow problems in terms of divergence-free functions can also be found in [15].…”

Section: Introductionmentioning

confidence: 98%

Direct numerical simulation of pipe flow using a solenoidal spectral method

Tuğluk

Tarman

2012

Acta Mech

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In this study, a numerical method based on solenoidal basis functions, for the simulation of incompressible flow through a circular-cylindrical pipe, is presented. The solenoidal bases utilized in the study are formulated using the Legendre polynomials. Legendre polynomials are favorable, both for the form of the basis functions and for the inner product integrals arising from the Galerkin-type projection used. The projection is performed onto the dual solenoidal bases, eliminating the pressure variable, simplifying the numerical approach to the problem. The success of the scheme in calculating turbulence statistics and its energy conserving properties is investigated. The generated numerical method is also tested by simulating the effect of drag reduction due to spanwise wall oscillations.

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“…Techniques of spectral methods are employed in leading to efficient implementation; however, the incorporation of the associated weight leads to added effort in the construction of the dual solenoidal bases. An analysis on the use of solenoidal bases in the numerical approximation of the Stokes problem is presented in ) and a domain decomposition procedure is proposed in to introduce some flexibility in the use of solenoidal bases. Other solenoidal bases are constructed empirically from flow database and used for optimal truncated representation of the underlying flow field such as Karhunen–Loeve bases .…”

Section: Introductionmentioning

confidence: 99%

Numerical simulations of thermal convection under the influence of an inclined magnetic field by using solenoidal bases

Yarımpabuç

Tarman

Yıldırım

2014

Math Methods in App Sciences

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The effect of an inclined homogeneous magnetic field on thermal convection between rigid plates heated from below under the influence of gravity is numerically simulated in a computational domain with periodic horizontal extent. The numerical technique is based on solenoidal (divergence-free) basis functions satisfying the boundary conditions for both the velocity and the induced magnetic field. Thus, the divergence-free conditions for both velocity and magnetic field are satisfied exactly. The expansion bases for the thermal field are also constructed to satisfy the boundary conditions. The governing partial differential equations are reduced to a system of ordinary differential equations under Galerkin projection and subsequently integrated in time numerically. The projection is performed by using a dual solenoidal bases set such that the pressure term is eliminated in the process. The quasi-steady relationship between the velocity and the induced magnetic field corresponding to the liquid metals or melts is used to generate the solenoidal bases for the magnetic field from those for the velocity field. The technique is validated in the linear case for both oblique and vertical case by reproducing the marginal stability curves for varying Chandrasekhar number. Some numerical simulations are performed for either case in the nonlinear regime for Prandtl numbers Pr D 0.05 and Pr D 0.1.

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Domain decomposition for spectral approximation to stokes equations via divergence-free functions

Cited by 4 publications

References 6 publications

Numerical Approximation of Partial Differential Equations

Numerical Approximation of Partial Differential Equations

Direct numerical simulation of pipe flow using a solenoidal spectral method

Numerical simulations of thermal convection under the influence of an inclined magnetic field by using solenoidal bases

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