In this article, we study some anisotropic singular perturbations for a class of linear elliptic problems. A uniform estimates for conforming Q1 finite element method are derived, and some other ...results of convergence and regularity for the continuous problem are proved.
This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both ...methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable
u
=
ψ
(
v
), where
ψ
is a well chosen diffeomorphism between (−1, 1) and ℝ, and
v
is valued in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D and 3D cases where the solution does not belong to
H
1
(Ω), show that this method can provide accurate results. We then construct a numerical scheme which presents a convergence property to the entropy weak solution of the problem in the case where the right-hand side belongs to
L
1
. This is achieved owing to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation, and adding an upstream weighting evaluation and a nonlinear
p
-Laplace vanishing stabilisation term.
We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution.
The ...velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties.
Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation.
Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates.
It is then possible to show the convergence of the approximate solution to a weak solution of the problem.
The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element.
We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan ...equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weighted p-Laplace term. A well chosen diffeomorphism between R and (−1,1) is used for the estimates of the approximated solution, and is involved in the above weight. We show that this approximation converges to a weak sense of the problem for general right-hand-side, and to the entropy solution in the case where the right-hand-side is in L1.
We prove existence of a solution to the implicit MAC scheme for the compressible Navier–Stokes equations. We derive error estimates for this scheme on two and three dimensional Cartesian grids. Error ...estimates are obtained by using the discrete version of the
relative energy method
introduced on the continuous level in Feireisl et al. (J Math Fluid Mech 14(4):717–730,
2012
). A systematic use of the theoretical “continuous” analysis of the equations in combination with the numerical tools is crucial for the result. This error estimate does not uses stability hypotheses on the solution of the numerical scheme.
Abstract
We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an ...exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.
The present paper addresses the convergence of a first-order in time incremental projection scheme for the time-dependent incompressible Navier–Stokes equations to a weak solution. We prove the ...convergence of the approximate solutions obtained by a semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield their existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.
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