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A method to approximate the Euler equations is presented. The method is a multi-domain approximation, and a variational form of the Euler equations is found by making use of the divergence theorem. The method is similar to that of the Discontinuous-Galerkin method of Cockburn and Shu, but the implementation is constructed through a spectral, multi-domain approach. The method is introduced and is shown to be a conservative scheme. A numerical example is given for the expanding flow around a point...
The newly developed unifying discontinuous formulation named the correction procedure via
reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes
equations for 3D mixed grids. In the current development, tetrahedrons and triangular
prisms are considered. The CPR method can unify several popular high order methods
including the discontinuous Galerkin and the spectral volume methods into a more efficient
differential form....
We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.
We analyze a new formulation of the Stokes equations in
three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to
the angular variable: the problem for each Fourier coefficient is two-dimensional and has
six scalar unknowns, corresponding to the vector potential and the vorticity. A
spectral discretization is built on this formulation, which leads to an exactly
divergence-free discrete velocity. We prove optimal error estimates.
The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.
The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.
The reduced basis element method is a new approach for approximating
the solution of problems described by partial differential equations.
The method takes its roots in domain decomposition methods and
reduced basis discretizations. The basic idea is to first decompose
the computational domain into a series of subdomains that are deformations
of a few reference domains (or generic computational parts).
Associated with each reference domain are precomputed solutions
corresponding to the same...
We consider the flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with mixed boundary conditions. Since the boundary pressure can present high variations, the permeability of the medium also depends on the pressure, so that the model is nonlinear. A posteriori estimates allow us to omit this dependence where the pressure does not vary too much. We perform the numerical analysis of a spectral element discretization of the simplified model. Finally we propose a strategy...
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