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Finite element approximation of a Stefan problem with degenerate Joule heating

John W. Barrett, Robert Nürnberg (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a fully practical finite element approximation of the following degenerate system ∂ ∂ t ρ ( u ) - ∇ . ( α ( u ) ∇ u ) ∋ σ ( u ) | ∇ φ | 2 , ∇ . ( σ ( u ) ∇ φ ) = 0 subject to an initial condition on the temperature, u , and boundary conditions on both u and the electric potential, φ . In the above ρ ( u ) is the enthalpy incorporating the latent heat of melting, α ( u ) > 0 is the temperature dependent heat conductivity, and σ ( u ) ≥ 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0 , which gives rise to the degeneracy in this Stefan system. In addition to showing stability...

Finite element approximation of a Stefan problem with degenerate Joule heating

John W. Barrett, Robert Nürnberg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a fully practical finite element approximation of the following degenerate system ∂ ∂ t ρ ( u ) - ∇ . ( α ( u ) ∇ u ) ∋ σ ( u ) | ∇ φ | 2 , ∇ . ( σ ( u ) ∇ φ ) = 0 subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan...

Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

John W. Barrett, Linda El Alaoui (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy...

Finite-volume level set method and its adaptive version in completing subjective contours

Zuzana Krivá (2007)

Kybernetika

In this paper we deal with a problem of segmentation (including missing boundary completion) and subjective contour creation. For the corresponding models we apply the semi-implicit finite volume numerical schemes leading to methods which are robust, efficient and stable without any restriction to a time step. The finite volume discretization enables to use the spatial adaptivity and thus improve significantly the computational time. The computational results related to image segmentation with partly...

Front propagation for nonlinear diffusion equations on the hyperbolic space

Hiroshi Matano, Fabio Punzo, Alberto Tesei (2015)

Journal of the European Mathematical Society

We study the Cauchy problem in the hyperbolic space ℍ n ( n ≥ 2 ) for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space ℝ n new phenomena arise, which depend on the properties of the diffusion process in ℍ n . We also investigate a family of travelling wave solutions, named...

Full regularity of bounded solutions to nondiagonal parabolic systems of two equations

Dmitry Portnyagin (2008)

Applicationes Mathematicae

Hölder continuity and, basing on this, full regularity and global existence of weak solutions is studied for a general nondiagonal parabolic system of nonlinear differential equations with the matrix of coefficients satisfying special structure conditions and depending on the unknowns. A technique based on estimating a certain function of unknowns is employed to this end.

Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

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