A key characteristic of mathematics is, that complicated concepts and relationships can be expressed by use of notations, similar to a language. This elegance of abstraction is especially vivid in finite element methods (FEM). FEM are frequently used for spatial discretisation of partial differential equations (PDEs), which can for example model physical processes. Firedrake  exploits this abstraction to encapsulate mathematical and computational details of FEM in order to make the method accessible for scientists, who are interested in its application to problems for example in the area of fluid dynamics.
The complexity of the underlying physical problems of PDEs translates into the numerical methods and requires a high amount of sophistication in order to achieve satisfying accuracy with an acceptable amount of resources. A concrete example for this is weather and climate prediction, where quantities have to be calculated on not only complex, but also massive domains and over various time scales. Influencing factors on the performance of these models can be of physical, mathematical or computational nature: examples are appropriate conversation of physical quantities, choice of spatial and temporal discretisation, and scalability . An implication of this is, that improvements on the numerical model require different researchers' expertise, which underpins the need of abstractions for certain subareas.
With my research I intend to contribute performance improvements to numerical models, in particular of high order FEM discretisation for PDEs. Discretisations as well as optimisations of the computations shall be made easily accessible for other scientists through automatic code generation.
Kleefeld A, Vorderwuelbecke S, Burgeth B, 2018, Anomalous diffusion, dilation, and erosion in image processing, International Journal of Computer Mathematics, Vol:95, ISSN:0020-7160, Pages:1375-1393
Vorderwuelbecke S, 2019, A Higher Order Mixed Discontinuous Finite Element Method For Incompressible Flows
Vorderwuelbecke S, 2018, Numerical Solutions of Fractional Nonlinear Advection-Reaction-Diffusion Equations