Research within the Numerical Analysis and Scientific Computing (NASC) group concentrates on the modelling and analysis of problems coming from engineering, the physical and the life sciences. The interests of the group range from rigorous existence and uniqueness analysis of (coupled systems of) PDEs and variational inequalities, through the development and analysis of numerical methods to the implementations of these methods (scientific computing). Current research directions comprise free boundary problems, numerical methods for nonlinear partial differential equations, optimisation with PDE constraints, inverse problems, large scale scattered data approximation and numerical methods for high dimensional problems. NASC has leading experts in both finite element and meshfree discretisation methods. Some examples of current projects are listed below.
- Faculty: Bertram Düring, Max Jensen, Omar Lakkis, Babis Makridakis, and Vanessa Styles.
- Postgraduate Students: Muhammad Abdullahi Yau, Maryam Asgir, Christof Heuer, Elham Khairi, Gary Poole, Giles Smith, Wanyun Zheng
Examples of current research projects
Fluid-Structure Interaction (FSI) deals with the impact of fluid flow on an elastic structure, which is either surrounded by the flow or filled with the fluid. Examples are in Aeroelasticity, where the structural deformations of aircrafts during flight are investigated, and in modelling cardiovascular flow. Mathematically, FSI requires the coupling of partial differential equations from fluid dynamics and structural mechanics. Members of the group have worked together with colleagues from EADS, Airbus, DLR Institutes and INRIA.
Research interests are in developing and analysing efficient methods for large-scale scattered data approximation, including high-dimensional data. Methods of interest comprise particularly radial basis functions, moving least squares, domain decomposition and partition of unity methods. Applications are in computer graphics, learning theory and meshfree methods for partial differential equations.
Free boundary problems
Research concentrates on the analysis and numerical treatment of free boundary problems motivated by shape and topology optimization, parameter estimation and control of differential variational inequalities. Besides the introduction of level-set based shape optimization methods, this led to the development of semismooth Newton techniques with many application areas ranging from contact problems in elasticity through computational finance to mathematical image processing.
Curvature dependent flow / diffusion induced grain boundary motion
The phenomenon of diffusion induced grain boundary motion (DIGM) can be observed if a polycrystalline metallic film is placed in a vapour containing another metal. Atoms from the metal diffuse into the film along the grain boundaries separating the crystals causing them to move – this motion is known as DIGM. The mathematical problem of DIGM is forced mean curvature flow with the forcing determined by the solution of another coupled elliptic/parabolic equation on the interface.
Constructive approximation on the sphere
Research concentrates on constructive methods for modelling phenomena on the sphere. In particular, efficient global and local quadrature rules are developed and investigated, but the research also encompasses radial basis functions and hyperinterpolation on the sphere. Past work includes the mathematical modelling for solving the inverse problem to reconstruct the earth's geopotential from satellite data.
Finite element methods for complex flow problems
Research comprises, for example, a priori and a posteriori error estimates for stabilised finite element methods using continuous or discontinuous approximation spaces, the design of monotone finite element methods and the analysis of the dissipative structure of continuous or discontinuous Galerkin methods.
Nonlinear parabolic equations and stochastic PDEs
Particular interest lies in geometrically based motions arising from free boundary problems; numerical methods for stochastic PDEs in phase transition. A posteriori error estimates are studied.