# Finite Element Approximation of the Navier-Stokes Equations ## Finite element approximation of steady Navier-Stokes equations with mixed boundary conditions

You can start or join in a discussion here. Visit emeraldpublishing. Abstract Purpose The purpose of this paper is to apply the variational multi-scale framework to the finite element approximation of the compressible Navier-Stokes equations written in conservation form. Findings Subsonic and supersonic numerical experiments show that including the orthogonal, dynamic, and the non-linear subscales improve the accuracy of the compressible formulation. Please note you might not have access to this content. You may be able to access this content by login via Shibboleth, Open Athens or with your Emerald account.

## Variational multi-scale finite element approximation of the compressible Navier-Stokes equations

William R. Schaum's Outline of Vector Analysis, 2ed. Murray R. A Student's Guide to Fourier Transforms.

Calculus Late Transcendentals. Howard Anton. An Introduction to Manifolds. We present some numerical tests with non-constant bottom, with the aim of showing, in practice, some of the advantages of the hydrostatic Stokes scheme Specifically, we exploit its flexibility for choosing different domains, which are approximated by standard finite element meshes not necessary structured. Convenient techniques, such as mesh adaptivity, are applied without additional difficulties.

Figures 13a and b show the meshes obtained after four solving plus mesh adaptation steps, using P2, P1 — P0 and P2, P1 — P1, respectively. Figures 13c-f show the velocity field and pressure isolines.

As we expected, both of them are, qualitatively, suitable results. Moreover, they agree with similar tests given in literature, where reduced integro-differential formulations in structured meshes are employed see. Specifically, two large vortices, one on each side of the interior mountain, are obtained, while pressure is constant in depth, increasing horizontally faster at the mountain peak of the bottom.

Note that pressure graphic for P2, Pi - Pi is better presents less oscillations than for P2, Pi - Po, needing the latter one a higher mesh refinement hence a higher computational effort. As counterpart, the velocity seems better for P2, Pi - P0. Note that, as predicted in Sect.

In fact, as reflected by Fig. Observe that one only vortex is obtained and extreme values ofpressure are concentrated near the "coast", where depth vanishes. Note also that it is not usual to find these kinds of experiments in literature, due to the difficulties presented by reduced integro-differential formulations when the domain has no talus. Siam J. I Math. Besson, O. Cushman-Roisin, B. Academic Press, New York Paris Srie I , Ciarlet, P.

North Holland, Amsterdam Cao, C. Integral Equ. Numerische Mathematik , Kimmritz, M. Lions, J. Nonlinearity 5, Nonlinearity 5, ii i Ortegon-Gallego, F. Nonlinear Anal. Theory Methods Appl. Pironneau, O. Stenberg, R. Methods Fluids 11, i Temam, R. American Mathematical Society, New York i Milan J.

In: Friedlander, Serre, D. Handbook of Mathematical Fluid Dynamics, vol.

Thin layer Navier Stokes approximation in three dimensions

Elsevier, New York Analysis of the hydrostatic Stokes problem and finite-element approximation in unstructured meshes Academic research paper on " Mathematics ". Similar topics of scientific paper in Mathematics , author of scholarly article — F. DOI Numerical tests The inf-sup condition IS On the other hand, we show that the approximation of u,v,p by P2, P1 — P0 which is Stokes-stable, see  is also hydrostatic-stable and we approach numerically its accuracy order in space.

1. A Finite Element Approximation of the Stokes Equations.
2. Finite Element Approximation of the Navier-Stokes Equations..
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Jz J-D x The existence of weak solutions of was obtained in . From this point of view, the original local hydrostatic Navier-Stokes problem has the advantage of allowing one to approximate by standard unstructured finite element meshes and to use a mesh adaptive tool. Theorem 3 The following statements are equivalent: 1. Proof We split the proof into four steps. Step 1. Rewriting the variational formulation Proof of the abstract inf-sup condition Step 3.

Proof of stability estimates Step 4. The reciprocal argument. Corollary 5 One has existence and uniqueness of solution of in W x P. Theorem 6 The following statements are equivalent: 1. Scheme 37 is well-posed in Xh. Proof Analogously to the first part of Lemma 1. Using that, it can be proved that P1,b, P1 — P0 is not Fig. Under some geometrical assumptions for the mesh family Th, a sufficient condition for P1,b, P1 — P1 to be Stokes-stable is : Th can be covered by a family of macroelements, Mh, verifying the following conditions: 1.

The macroelements in Mh contain one only interior element. Oscillations can be observed in pressure but not in velocity Fig. References 1. Brezzi, F. Springer, New York 5. Academic Press, New York 7. Paris Srie I , 8. Ern, A. Springer, Berlin Finite Element Approximation of the Navier-Stokes Equations Finite Element Approximation of the Navier-Stokes Equations Finite Element Approximation of the Navier-Stokes Equations Finite Element Approximation of the Navier-Stokes Equations Finite Element Approximation of the Navier-Stokes Equations Finite Element Approximation of the Navier-Stokes Equations Finite Element Approximation of the Navier-Stokes Equations Finite Element Approximation of the Navier-Stokes Equations
Finite Element Approximation of the Navier-Stokes Equations

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