Interfaces and free boundary problems are ubiquitous in the mathematical modelling of physical phenomena arising, for example, as material interfaces in fluids and solids and as phase boundaries in alloys. Newer applications are in, for example, geometrically based models for image processing. Because of their inherent nonlinearity and the difficulties of dealing with unknown and moving curved surfaces on fixed grids they are well known to be computationally challenging. Specific application areas frequently need their own mathematical modelling and analysis, but also can be tackled computationally by general approaches such as front tracking, the immersed interface method, phase field equations and the level set method. In order to gain full understanding of the models, one needs to rely on mathematical analysis and scientific computation. It is increasingly clear that efficient and successful numerical discretisation requires consideration of the modelling as well as of the numerical analysis. Complicated behaviour of interfaces is common as a consequence of instability, nonlinear degeneracy, the interaction of scales and physics in the bulk with physics on the surface. The analysis of discretisation of these problems as well as the implementation of schemes is demanding.
There have been spectacular advances recently in the application of level set methods, parametric formulations and phase field equations. The mathematical analysis of these methods is maturing but there are many open problems with respect to error bounds and convergence for geometrical motion, coupled to field equations. Topological change due to pinching off and merging of interfaces remain one of the most computationally challenging problems in partial differential equations. The question of which computational approach is best remains controversial.
The purpose of this workshop is to highlight the important recent advances associated with the mathematical analysis of computational methods and the development of computational tools. The latest developments in mathematical models and the computational challenges they present will also be addressed.
The meeting will have three inter-related themes:
- Models and Applications: Phase transformations and solidification, grain boundary motion, superconductivity, thin films, microstructure, geometrical motion, image processing, material interfaces in solids and fluids.
- Methodology and analysis: Finite element methods, level set methods, phase field and front tracking methods, analysis of discretisations.
- Simulation: Engineering and scientific computations, visualisation, comparison of methodologies.