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A weak solution of the coupling of time-dependent incompressible Navier–Stokes equations
with Darcy equations is defined. The interface conditions include the
Beavers–Joseph–Saffman condition. Existence and uniqueness of the weak solution are
obtained by a constructive approach. The analysis is valid for weak regularity
We consider the flow of a viscous incompressible fluid through a rigid
homogeneous porous medium. The permeability of the medium depends
on the pressure, so that the model is nonlinear. We propose a finite
element discretization of this problem and, in the case where the
dependence on the pressure is bounded from above and below, we prove
its convergence to the solution and propose an algorithm to solve
the discrete system. In the case where the dependence
on the pressure is exponential, we propose a splitting
scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.
In this paper we solve the time-dependent incompressible Navier-Stokes
equations by splitting the non-linearity and incompressibility, and
using discontinuous or continuous finite element methods in space. We
prove optimal error estimates for the velocity and suboptimal
estimates for the pressure. We present some numerical experiments.
We propose and analyze several finite-element schemes for solving a grade-two
fluid model, with a
tangential boundary condition, in a two-dimensional polygon. The exact
problem is split into a
generalized Stokes problem and a transport equation, in such a way that it
always has a solution
without restriction on the shape of the domain and on the size of the data.
The first scheme uses
divergence-free discrete velocities and a centered discretization of the
transport term, whereas the
other schemes use Hood-Taylor discretizations for the velocity and
pressure, and either a centered or an upwind
discretization of the transport term. One facet of our analysis is
that, without restrictions on the data,
each scheme has a discrete solution and all discrete solutions converge
strongly to solutions of the
exact problem. Furthermore, if the domain is convex and the data satisfy
certain conditions, each
scheme satisfies error inequalities that lead to error estimates.
We semi-discretize in space a time-dependent Navier-Stokes system
on a three-dimensional polyhedron by finite-elements schemes
defined on two grids. In the first step, the fully non-linear
problem is semi-discretized on a coarse grid, with mesh-size H.
In the second step, the problem is linearized by substituting
into the non-linear term, the velocity uH computed at step
one, and the linearized problem is semi-discretized on a fine
grid with mesh-size h. This approach is motivated by the fact
that, on a convex polyhedron and under adequate assumptions on the
data, the contribution of uH to the error analysis is
measured in the L2 norm in space and time, and thus, for the
lowest-degree elements, is of the order of H2. Hence, an error
of the order of h can be recovered at the second step, provided
h = H2.
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