Minisymposium
at EQUADIFF 11,
Bratislava

**Nonlinear Diffusion and Motion of
Interfaces**

organized by Michal Beneš

**Participants:**

Kenji TOMOEDA (Osaka Institute of Technology, Osaka, Japan)

Tatsuyuki NAKAKI (Kyushu University, Fukuoka, Japan)

Yohei KASHIMA (University of Sussex, Brighton, UK)

Michal BENEŠ (Czech Technical University in Prague, CZ)

**Titles
and
abstracts:**

The support re-splitting phenomena caused by the interaction between

diffusion and absorption by K. Tomoeda

Abstract:

Numerical experiments suggest the interesting properties in the several fields. One of them is the occurrence of support*re-splitting
phenomena*
caused by the interaction between diffusion and absorption, where
support
splitting phenomena means that the region occupied by the flow becomes
disconnected. From only numerical computations it is difficult to
justify whether such phenomena
are true or not, because the space mesh and the time step are
sufficiently small but not zero. So the mathematical analysis is
needed. In this talk such phenomena are investigated by use of
finite difference scheme, and justified from numerical and analytical
points of view. The interface equation, the comparison theorem and the
nonincrease of the number of local maximum points for the solution are
used.

Approximations for some diffusion and interface

problems using singular limit technique by T. Nakaki and H. Murakawa

Abstract:

Approximations, which are useful and effective for numerical simulations, are proposed. The approximations are constructed by using singular limit solutions of certain reaction-diffusion equations. The approximations are constructed by using singular limit solutions of certain reaction-diffusion equations. First example is Stefan problem. We can easily and clearly capture the interfaces in multi-dimensional space by using the approximation. Another example is a nonlinear diffusion equation. The approximation also works well, and we obtain good numerical solution with comparatively low computational cost. In this talk, we discuss the approximations and demonstrate some numerical simulations.

A finite element analysis of macroscopic models for superconductivity in 3D. by Y. Kashima

Abstract:

A finite element analysis of macroscopic models for type II superconductor in 3D configuration is considered. We formulate the magnetic field induced by the supercurrent flowing though the bulk superconductor in an evolution variational inequality. Introducing a magnetic scalar potential and a penalized energy leads to an unconstrained minimization problem. The discretization is carried out by employing curl conforming edge element. We argue the convergence property to the analytical solution by passing mesh size, time step and penalty coefficient to zero. Some numerical results will be reported.

Quantitative aspects of microstructure formation in solidification by M. Beneš

Abstract:

The growth of microstructure non-convex patterns is studied by means of the modified anisotropic phase-field model. The numerical algorithm is designed using the finite-difference spatial discretisation in the method of lines. Beside the numerical analysis of the model which is using the a-priori estimates and the compactness and monotonicity arguments, we present a series of qualitative studies demonstrating ability of the model. A special attention is paid to the implementation issues such as handling of high CPU-cost parts of the code and parallelization. As a quantitative result, we present the convergence studies when mesh size and diffuse parameter tend to zero.

organized by Michal Beneš

Kenji TOMOEDA (Osaka Institute of Technology, Osaka, Japan)

Tatsuyuki NAKAKI (Kyushu University, Fukuoka, Japan)

Yohei KASHIMA (University of Sussex, Brighton, UK)

Michal BENEŠ (Czech Technical University in Prague, CZ)

The support re-splitting phenomena caused by the interaction between

diffusion and absorption by K. Tomoeda

Abstract:

Numerical experiments suggest the interesting properties in the several fields. One of them is the occurrence of support

Approximations for some diffusion and interface

problems using singular limit technique by T. Nakaki and H. Murakawa

Abstract:

Approximations, which are useful and effective for numerical simulations, are proposed. The approximations are constructed by using singular limit solutions of certain reaction-diffusion equations. The approximations are constructed by using singular limit solutions of certain reaction-diffusion equations. First example is Stefan problem. We can easily and clearly capture the interfaces in multi-dimensional space by using the approximation. Another example is a nonlinear diffusion equation. The approximation also works well, and we obtain good numerical solution with comparatively low computational cost. In this talk, we discuss the approximations and demonstrate some numerical simulations.

A finite element analysis of macroscopic models for superconductivity in 3D. by Y. Kashima

Abstract:

A finite element analysis of macroscopic models for type II superconductor in 3D configuration is considered. We formulate the magnetic field induced by the supercurrent flowing though the bulk superconductor in an evolution variational inequality. Introducing a magnetic scalar potential and a penalized energy leads to an unconstrained minimization problem. The discretization is carried out by employing curl conforming edge element. We argue the convergence property to the analytical solution by passing mesh size, time step and penalty coefficient to zero. Some numerical results will be reported.

Quantitative aspects of microstructure formation in solidification by M. Beneš

Abstract:

The growth of microstructure non-convex patterns is studied by means of the modified anisotropic phase-field model. The numerical algorithm is designed using the finite-difference spatial discretisation in the method of lines. Beside the numerical analysis of the model which is using the a-priori estimates and the compactness and monotonicity arguments, we present a series of qualitative studies demonstrating ability of the model. A special attention is paid to the implementation issues such as handling of high CPU-cost parts of the code and parallelization. As a quantitative result, we present the convergence studies when mesh size and diffuse parameter tend to zero.