Finite element methods provide a versatile and mathematically well-founded approach to solve partial differential and integral equations arising from mathematical models in physics and engineering. The research activities in this field can be subdivided into:

### Development of new algorithms

*hp*-adaptive discontinuous Galerkin finite element methods for partial differential equations.Special attention is given to:

- Space-time discontinuous Galerkin finite element methods in order to deal with moving boundaries and deforming meshes
- Efficient multigrid and (pseudo)-time integration methods to solve the large systems of ordinary differential or nonlinear
algebraic equations resulting from DG discretizations
- Adaptation algorithms controlled by a posteriori error estimates

- Space-time discontinuous Galerkin finite element methods in order to deal with moving boundaries and deforming meshes
**Finite element discretizations which preserve important mathematical properties of the underlying partial differential equations**Special attention is given to:

- Coupled continuous-discontinuous Galerkin finite element methods
- Vector finite elements, such as Whitney elements

- Coupled continuous-discontinuous Galerkin finite element methods
**Galerkin least squares finite element methods**Special attention is given to:

- Development of stabilization operators with a solid mathematical background

**Theoretical analysis of finite element methods**Special attention is given to:

- Stability analysis
- A priori error analysis to investigate the optimal rate of convergence
- A posteriori error analysis to control adaptation algorithms

- Stability analysis
**Applications**Special attention is given to:

- Compressible, incompressible and (dispersed) multiphase flows, including problems with free surfaces
- Maxwell equations
- Development of a general purpose finite element toolkit
*hpGEM*to construct advanced simulation programs

- Compressible, incompressible and (dispersed) multiphase flows, including problems with free surfaces