For linear and non-linear partial differential equations (PDE) we investigate mesh-based methods ranging from continuous FEM (finite element) to DG (discontinuous Galerkin) methods. The main task is to estimate the discretisation error between exact and discrete solution. Such estimates ensure convergence of numerical schemes if the solutions are sufficiently regular. Key ingredients are stability properties of the numerical method and approximation properties of the underlying discrete spaces. Even though they occur in different shapes the main principles are the same independent of partial differential equation and numerical scheme. For example, stability results for incompressible fluid equations are based on the inf-sup stability, whereas for compressible fluid equations relative energy estimates on the discrete level represent a powerful tool.
We are also concerned with the development and analysis of time integrators to solve large-scale systems of ordinary differential equations arising, e.g., from semi-discretizations of PDEs. The main focus is to construct higher-order methods which are suitable for problems with different time scales and optimal control problems constrained by ordinary differential equations.
