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