This paper presents an extension to stabilized methods of the standard technique for the numerical analysis of mixed methods. We prove that the stability of stabilized methods follows from an underlying discrete inf-sup condition, plus a uniform separation property between bubble and velocity finite element spaces. We apply the technique introduced to prove the stability of stabilized spectral element methods so as stabilized solution of the primitive equations of the ocean.

This paper presents an
extension to stabilized methods of the standard technique for the
numerical analysis of mixed methods. We prove that the stability of stabilized
methods follows from an underlying discrete inf-sup condition, plus a uniform
separation property between bubble and velocity finite element spaces. We apply
the technique introduced to prove
the sta bi li ty of stabilized spectral element methods so as
stabilized solution of the primitive equations of the ocean.

In this work we introduce an accurate solver for the Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

This paper introduces a scheme for the numerical approximation of a
model for two turbulent flows with coupling at an interface. We
consider the variational formulation of the coupled model, where the
turbulent kinetic energy equation is formulated by transposition. We
prove the convergence of the approximation to this formulation for
3D flows for large turbulent viscosities and smooth enough
flows, whenever bounded in
Sobolev norms for large enough. Under the same assumptions,...

The hydrostatic approximation of the incompressible 3D stationary
Navier-Stokes equations is widely used in oceanography and other
applied sciences. It appears through a limit process due to
the anisotropy of the domain in use, an ocean, and it is usually studied as
such.
We consider in this paper an equivalent formulation to this
hydrostatic approximation that includes Coriolis force and an additional
pressure term that comes from taking into account the
pressure in the state equation for...

The hydrostatic approximation of the incompressible 3D stationary
Navier-Stokes equations is widely used in oceanography and other
applied sciences. It appears through a limit process due to
the anisotropy of the domain in use, an ocean, and it is usually studied as
such.
We consider in this paper an equivalent formulation to this
hydrostatic approximation that includes Coriolis force and an additional
pressure term that comes from taking into account the
pressure in the state equation for...

In this work we introduce an accurate solver for the
Shallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove and error estimates for the resulting discrete problem. Some numerical experiments confirm the interest...

We introduce in this paper some elements for the mathematical and numerical analysis of algebraic turbulence models for
oceanic surface mixing layers. In these models the turbulent diffusions are parameterized by means of the gradient Richardson number, that measures the balance between stabilizing buoyancy forces and destabilizing shearing forces. We analyze the existence and linear exponential asymptotic stability of continuous and discrete equilibria states. We also analyze the well-posedness...

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