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Numerical analysis of nonlinear elliptic-parabolic equations

Emmanuel Maitre (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

Numerical methods for fourth order nonlinear degenerate diffusion problems

Jürgen Becker, Günther Grün, Martin Lenz, Martin Rumpf (2002)

Applications of Mathematics

Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant),...

On an over-determined problem of free boundary of a degenerate parabolic equation

Jiaqing Pan (2013)

Applications of Mathematics

This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants K and T 0 , to decide the initial value u 0 such that the solution u ( x , t ) satisfies sup x H u ( T 0 ) | x | K , where H u ( T 0 ) = { x N : u ( x , T 0 ) > 0 } . In this paper, we first establish a priori estimate u t C ( t ) u and a more precise Poincaré type inequality...

On fully practical finite element approximations of degenerate Cahn-Hilliard systems

John W. Barrett, James F. Blowey, Harald Garcke (2001)

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

We consider a model for phase separation of a multi-component alloy with non-smooth free energy and a degenerate mobility matrix. In addition to showing well-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in order to be applicable to a logarithmic free energy. Finally numerical experiments with...

On fully practical finite element approximations of degenerate Cahn-Hilliard systems

John W. Barrett, James F. Blowey, Harald Garcke (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a model for phase separation of a multi-component alloy with non-smooth free energy and a degenerate mobility matrix. In addition to showing well-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in order to be applicable to a logarithmic free energy. Finally numerical experiments...

On higher-order semilinear parabolic equations with measures as initial data

Victor Galaktionov (2004)

Journal of the European Mathematical Society

We consider 2 m th-order ( m 2 ) semilinear parabolic equations u t = ( Δ ) m u ± | u | p 1 u in N × + ( p > 1 ) , with Dirac’s mass δ ( x ) as the initial function. We show that for p < p 0 = 1 + 2 m / N , the Cauchy problem admits...

On solutions of a perturbed fast diffusion equation

Ján Filo (1987)

Aplikace matematiky

The paper concerns the (local and global) existence, nonexistence, uniqueness and some properties of nonnegative solutions of a nonlinear density dependent diffusion equation with homogeneous Dirichlet boundary conditions.

Currently displaying 241 – 260 of 417