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The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading...
The Cahn-Hilliard variational inequality is a non-standard
parabolic variational inequality of fourth order for which
straightforward numerical
approaches cannot be applied. We propose a primal-dual active set
method which can be interpreted as a semi-smooth Newton method as
solution technique for the discretized Cahn-Hilliard variational
inequality. A (semi-)implicit Euler discretization is used in time
and a piecewise linear finite element discretization of splitting
type is used in space...
A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points...
We show that the classical solution of the heat equation can be seen as the minimizer of a suitable functional defined in space-time. Using similar ideas, we introduce a functional on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with . We show some connections between minimizers of and mean curvature flow.
We consider three types of semilinear second order PDEs on a cylindrical domain , where is a bounded domain in , . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of is reserved for time , the third type is an elliptic equation with a singled out unbounded variable . We discuss the asymptotic behavior, as , of solutions which are defined and bounded on .
The author proves the existence of the asymptotic error expansion to the Peaceman-Rachford finite-difference scheme for the first boundary value problem of the two-dimensional evolationary equation on the so-called uniform and nearly uniform domains. This expansion leads, by Richardson extrapolation, to a simple process for accelerating the convergence of the method. A numerical example is given.
This survey is devoted to some fractional extensions of the incomplete
lumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems for
the fractional heat equation. By using Caputo’s differintegration operator
and the Laplace transform, new integral forms of the solutions are obtained.
In each of the different cases the integrands are expressed...
En esta nota se analizan dos modelos matemáticos deterministas planteados en problemas ecológicos causados por la introducción de nuevas especies en ambientes insulares heterogéneos. En el primero desarrollamos un modelo epidemológico con transmisión indirecta del virus por medio del ambiente. En el segundo se introduce un modelo específico de depredador-presa que exhibe la extinción en tiempo finito de las especies. Ambos modelos involucran sistemas de ecuaciones en derivadas parciales con interesantes...
We derive in this article some models of
Cahn-Hilliard equations in nonisotropic media. These models, based on
constitutive equations introduced by Gurtin in [19], take the work of
internal microforces and also the deformations of the material into
account. We then study the existence and uniqueness of solutions and
obtain the existence of finite dimensional attractors.
We give examples of parabolic systems (in space dimension ) having a solution with real analytic initial and boundary values which develops the discontinuity in the interior of the parabolic cylinder.
An existence and uniqueness theorem for a nonlinear parabolic system of partial differential equations, connected with the theory of heat conduction with a transition phase in a concentrated capacity, is given in sufficiently general hypotheses on the data.
We address three null controllability problems related to the heat equation. First we show that the heat equation with a rapidly oscillating density is uniformly null controllable as the period of the density tends to zero. We also prove that the same result holds for the finite-difference semi-discretization in space of the constant coefficient heat equation as the step size tends to zero. Finally, we prove that the null controllability of the constant coefficient heat equation can be obtained...
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