Discontinuous Galerkin Finite Elements for Moving Boundary Problems  

Project description

Many physical and chemical processes, typified by those related to fluid flow, can be modelled mathematically using partial differential equations. These can usually only be solved in the simplest of situations, but solutions in far more complex cases can be approximated using numerical and computational techniques. Traditional approaches to providing these computational simulations have typically modelled the evolution of the system by approximating the equations on a uniform mesh of points covering a domain with a fixed boundary. However, many situations (consider the spreading of a droplet, for example), naturally suggest a domain which evolves with the flow, while the main focus of interest in others (say the movement of a shock wave up and down an aeroplane wing) is in following the motion of a sharp internal feature. For accuracy and efficiency a computational method should not only approximate the partial differential equations appropriately, but also move the computational mesh in a manner which follows such features.

Recent research has developed a finite element approach to the adaptive approximation of time-dependent physical problems involving moving boundaries or interfaces. It has been deliberately designed to preserve inherent properties (such as conservation principles and invariances) of the underlying partial differential equations and hence of the system the mathematics is intended to represent. Extremely promising results have been obtained for a wide range of problems in one and two space dimensions, but the applicability of the approach is still limited (as are all moving mesh methods) by the potential for the computational mesh to "tangle".

The aim of this project will be to develop an alternative approach, derived within the same framework, which takes advantage of the additional flexibility inherent in the discontinuous Galerkin finite element framework. This has the potential to reduce the occurrence of mesh tangling and to greatly improve the robustness of the method when modelling problems involving complex, interacting features and when using different monitor to govern the movement of the mesh.