283 research outputs found
Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes
We study a colocated cell centered finite volume method for the approximation
of the incompressible Navier-Stokes equations posed on a 2D or 3D finite
domain. The discrete unknowns are the components of the velocity and the
pressures, all of them colocated at the center of the cells of a unique mesh;
hence the need for a stabilization technique, which we choose of the
Brezzi-Pitk\"aranta type. The scheme features two essential properties: the
discrete gradient is the transposed of the divergence terms and the discrete
trilinear form associated to nonlinear advective terms vanishes on discrete
divergence free velocity fields. As a consequence, the scheme is proved to be
unconditionally stable and convergent for the Stokes problem, the steady and
the transient Navier-Stokes equations. In this latter case, for a given
sequence of approximate solutions computed on meshes the size of which tends to
zero, we prove, up to a subsequence, the -convergence of the components of
the velocity, and, in the steady case, the weak -convergence of the
pressure. The proof relies on the study of space and time translates of
approximate solutions, which allows the application of Kolmogorov's theorem.
The limit of this subsequence is then shown to be a weak solution of the
Navier-Stokes equations. Numerical examples are performed to obtain numerical
convergence rates in both the linear and the nonlinear case.Comment: submitted September 0
The gradient discretisation method for linear advection problems
We adapt the Gradient Discretisation Method (GDM), originally designed for
elliptic and parabolic partial differential equations, to the case of a linear
scalar hyperbolic equations. This enables the simultaneous design and
convergence analysis of various numerical schemes, corresponding to the methods
known to be GDMs, such as finite elements (conforming or non-conforming,
standard or mass-lumped), finite volumes on rectangular or simplicial grids,
and other recent methods developed for general polytopal meshes. The scheme is
of centred type, with added linear or non-linear numerical diffusion. We
complement the convergence analysis with numerical tests based on the
mass-lumped P1 conforming and non conforming finite element and on the hybrid
finite volume method
A unified analysis of elliptic problems with various boundary conditions and their approximation
We design an abstract setting for the approximation in Banach spaces of
operators acting in duality. A typical example are the gradient and divergence
operators in Lebesgue--Sobolev spaces on a bounded domain. We apply this
abstract setting to the numerical approximation of Leray-Lions type problems,
which include in particular linear diffusion. The main interest of the abstract
setting is to provide a unified convergence analysis that simultaneously covers
(i) all usual boundary conditions, (ii) several approximation methods. The
considered approximations can be conforming, or not (that is, the approximation
functions can belong to the energy space of the problem, or not), and include
classical as well as recent numerical schemes. Convergence results and error
estimates are given. We finally briefly show how the abstract setting can also
be applied to other models, including flows in fractured medium, elasticity
equations and diffusion equations on manifolds. A by-product of the analysis is
an apparently novel result on the equivalence between general Poincar{\'e}
inequalities and the surjectivity of the divergence operator in appropriate
spaces
A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
We investigate the connections between several recent methods for the
discretization of anisotropic heterogeneous diffusion operators on general
grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite
Volume scheme and the Mixed Finite Volume scheme are in fact identical up to
some slight generalizations. As a consequence, some of the mathematical results
obtained for each of the method (such as convergence properties or error
estimates) may be extended to the unified common framework. We then focus on
the relationships between this unified method and nonconforming Finite Element
schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit
lifting operator close to the ones used in some theoretical studies of the
Mimetic Finite Difference scheme. We also show that for isotropic operators, on
particular meshes such as triangular meshes with acute angles, the unified
method boils down to the well-known efficient two-point flux Finite Volume
scheme
The gradient discretisation method for linear advection problems
International audienceWe adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped P1 conforming and non conforming finite element and on the hybrid finite volume method
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