Variational Space-Time Elements

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Discretisation in space using finite elements are widely used today. Efficient parallelization techniques have been developed throughout the past decades, based on suitable domain decomposition methods in space. Nevertheless, if subdomains becomes to small, scaling saturates especially for time dependent problems. To deal with this issue, a wide range of projects are working on temporal parallelization; however, the usage of time stepping algorithms is a drawback due to the restriction of the flow of informations in positive temporal direction.Moreover, these classical time stepping scheme negates one of the most powerful features of the finite element method: unstructured meshes. In this proposal, we aim at a direct finite element discretisation of the space-time using a continuous Bubnov-Galerkin approach, suitable for massive parallel analysis of the arising large-scale problem. This approach is in stark contrast to classical space-time formulation, which often use discontinuous Galerkin methods or a reduced order of the test or trial functions in time. Moreover, this approach allows us to deal with non-smooth systems and conservation properties in Hamiltonian systems.The construction of three- and fourdimensional space-time elements for non-linear elasticity as well as new solution techniques for contact constrains are in the main focus of this proposal. Novel multifield formulations allows to understand the differential-geometrical structure of the elastic system and to design suitable space-time elements.