Software for finite element methods and its application to nonvariational problems

We begin by introducing an extension to the software package Dune (a C++ based toolbox for solving PDEs with the finite element method) which has the main objective of providing a Python user interface to it. First of all we explain how we have structured the interface and go into some detail about the components typical to a FEM. We then go on to demonstrate different features available in the context of worked examples. For instance, we consider the integration of different software packages such as PETSc and SciPy, as well as FEM features such as grid adaptivity, moving domains, and partitioned grids. Throughout this we highlight design decisions that are different to other similar packages and the reasoning behind them. We conclude by demonstrating how C++ code development can be integrated into the process and how that affects efficiency. We go on to consider an application of this software to nonvariational PDEs. The key contribution of this section is the development of a new method for solving this class of problems based on minimization. We derive this method and provide results for existence and uniqueness and error convergence. We also compare this method to existing methods and highlight the advantages it has. We then derive a second aspect of this method which involves a finite element version of the Hessian. We combine these features and look at numerical results for linear nonvariational problems. We compare the new methods along with other existing methods using our software in terms of convergence rates and efficiency. Finally we take an experimental look at solving nonlinear nonvariational problems using the finite element Hessian, and an application to the Monge-Ampere equation

Mathematical Modelling of the Impact of Liquid Properties on Droplet Size from Flat Fan Nozzles

Flat fan nozzles atomize crop protection products, breaking them into droplets. Droplet size matters - smaller droplets give better perfor- mance, but very small droplets drift. We want to use mathematical models to better understand how liquid properties affect droplet size. There are three types of breakup: wavy sheet, perforation, and rim. In wavy sheet breakup, increasing viscosity or surface tension increases droplet size. To investigate further, we carry out direct numerical simulations of jet breakup, which show that suface tension has little effect, but increasing viscosity leads to fewer droplets. Decreasing the jet velocity also results in fewer droplets, with a wider size distribution. Each type of breakup involves primary breakup into cylinders of fluid, then secondary breakup into droplets. We thus consider the breakup of a cylinder of fluid. Direct numerical simulations suggest that within the tested parameter range viscosity has little impact on droplet size, however it does influence the timescale on which the instability evolves considerably. Linear stability analysis suggests that increasing viscosity increases the wavelength of the most unstable mode, which we expect leads to larger droplets, and that it reduces the rate of breakup. Perforations - holes in the sheet - also lead to breakup. We find how the length fraction of the sheet that is void changes with time. After breakup, the droplets continue to evolve. We develop a model, based on a transport equation, for this process. A key parameter is the breakup rate constant - larger values lead to more breakup, fewer large droplets, and a narrower size distribution. Together, these mathematical approaches improve our understanding of how droplets form, and can be used to guide experimental work