We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic ...p-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original problem as the perturbed data approach the original data. We do not assume uniqueness or regularity. When uniqueness is known, our result demonstrates that the weak solution is uniformly temporally stable to perturbations of the data. Beginning with a proof of temporally-uniform, spatially-weak convergence, we strengthen the latter by relating the unknown to an underlying convex structure that emerges naturally from energy estimates. The double degeneracy — shown to be equivalent to a maximal monotone operator framework — is handled with techniques inspired by a classical monotonicity argument and a simple variant of the compensated compactness phenomenon.
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.
Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming finite ...element, mixed finite element and finite volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards’, Stefan’s and Leray–Lions’ models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli–Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin–Simon theorem. The model’s degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.
Lions' representation theorem and applications Arendt, W.; Chalendar, I.; Eymard, R.
Journal of mathematical analysis and applications,
06/2023, Letnik:
522, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The Lions' Representation Theorem (LRT) is a version of the Lax–Milgram Theorem where completeness of one of the spaces is not needed. In this paper, LRT is deduced from an operator-theoretical ...result on normed spaces, which is of independent interest. As an example, we give a new characterization of dissipativity. The main part of the paper is a theory of derivations, based on LRT, which we develop. Its aim is to establish well-posedness results, not only for evolution in time but also for more general settings in terms of this new notion of derivation. One application concerns non-autonomous evolution equations with a new kind of boundary condition where values at the initial and final time are mixed.
![Convergence in C([0,T];L2(Ω...](http://api.summon.serialssolutions.com/2.0.0/image/issn/XR4PS5YY4K/0022-0396/small "Convergence in C([0,T];L2(Ω)) of weak solutions to perturbed doubly degenerate parabolic equations")
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