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We introduce and analyze families of Galerkin--collocation discretization schemes in time for the wave equation. Their conceptual basis is the establishment of a connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex algebraic systems. Firstly, continuously differentiable in time discrete solutions are studied. Optimal order error estimates are proved. Then, the concept of Galerkin--collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct link between the two families by a computationally cheap post-processing is presented. A key ingredient of the proposed methods is the application of quadrature rules involving derivatives. The performance properties of the schemes are illustrated by numerical experiments.

We propose and analyze numerically a new fictitious domain method, based on higher order space-time finite element discretizations, for the simulation of the nonstationary, incompressible Navier-Stokes equations on evolving domains. The physical domain is embedded into a fixed computational mesh such that arbitrary intersections of the moving domain's boundaries with the background mesh occur. The key ingredients of the approach are the weak formulation of Dirichlet boundary conditions by Nitsche's method, the flexible and efficient integration over all types of intersections of cells by moving boundaries and the spatial extension of the discrete physical quantities to the entire computational background mesh including fictitious (ghost) subdomains of fluid flow. To prevent spurious oscillations caused by irregular intersections of mesh cells, a penalization ensuring the stability of the approach and defining implicitly the extension to host domains is added. These techniques are embedded in an arbitrary order, discontinuous Galerkin discretization of the time variable and an inf-sup stable discretization of the spatial variables. The convergence and stability properties of the approach are studied, firstly, for a benchmark problem of flow around a stationary obstacle and, secondly, for flow around moving obstacles with arising cut cells and fictitious domains. The parallel implementation is also addressed.

The elucidation of many physical problems in science and engineering is subject to the accurate numerical modelling of complex wave propagation phenomena. Over the last decades, high-order numerical approximation for partial differential equations has become a well-established tool. Here we propose and study numerically the implicit approximation in time of wave equations by a Galerkin-collocation approach that relies on a higher order space-time finite element approach. The conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. For the fully discrete solution, higher order regularity in time is further ensured which can be advantageous in the discretization of multi-physics systems. The accuracy and efficiency of the variational collocation approach is carefully studied by numerical experiments.