Research

Space-time finite element approximation schemes offer appreciable advantages over the more standard discretization schemes combining finite element techniques for the discretization of the spatial variables with finite difference methods for the discretization of the temporal variable. Space-time finite element methods

offer the natural construction of higher order schemes within their families,
lead to uniform variational space-time formulations being advantageous for their analysis,
allow the construction of discontinuous, continuous and Cr-regular families (with respect to the time variable) of discretization schemes with different stability and convergence properties, such that the choice of the scheme can be based on the underlying mathematical problem (for instance, parabolic or 2nd order hyperbolic) and solution’s properties and regularity, and
let error control mechanism and adaptive techniques, for instance Dual Weighted Residual error control and hp adaptivity, being developed for the space discretization, become applicable for the space-time error control and adaptivity.
Great potential is further obtained for the numerical approximation of complex coupled problems of multiphysics, the development of multiscale methods for the discretization in time and parallel-in-time integration for large scale computations on high performance computer systems.  The efficient solution of the arising high dimensional algebraic systems continues to remain a field of active research.

By the group several families of space-time finite element techniques have been analyzed and implemented in the group’s parallel frontend software DTM++ to the deal.II library. Elastic wave propagation phenomena in composites and fibre-reinforced polymers with application in structural health monitoring and non-destructive material inspection as well as fluid-structure interaction in poroelasticity have been studied thoroughly. In particular, the group’s methods and results have been published in the following works.

• M. P. Bruchhäuser, U. Köcher, M. Bause, On the implementation of an adaptive multirate framework for coupled transport and flow, J. Sci. Comput. 93, 59 (2022); [https://doi.org/10.1007/s10915-022-02026-z]  [arXiv:2202.03748v1]
• M. Bause, U. Köcher, F. A. Radu, F. Schieweck, Post-processed Galerkin approximation of improved order for wave equations, Math. Comp., submitted (2018), pp. 1–34; arXiv:1803.03005
• U. Köcher, M. Bause, A mixed discontinuous-continuous Galerkin times discretisation for Biot’s system, Adv. Comp. Math., submitted (2018), pp. 1–19; arXiv:1805.00771
• M. Bause, Iterative coupling of mixed and discontinuous Galerkin methods for poroelasticity, Numer. Math. Adv. Appl. ENUMATH 2017, Springer, Berlin, 2017, accepted, pp. 1–8; arXiv:1802.03230
• U. Köcher, Influence of the SIPG penalisation on the numerical properties of linear systems for elastic wave propagation, Numer. Math. Adv. Appl. ENUMATH 2017, Springer, Berlin, 2017, accepted, pp. 1–8; arXiv:1712.05594
• J. Both, U. Köcher, Numerical investigation on the fixed-stress splitting scheme for Biot’s equations: Optimality of the tuning parameter, Numer. Math. Adv. Appl. ENUMATH 2017, Springer, Berlin, 2017, accepted, pp. 1–8; arXiv:1801.08352
• M. Bause, R. Radu, U. Köcher, Space-time finite element approximation of the Biot poroelasticity system with iterative coupling, Comput. Meth. Appl. Mech. Engrg., 320 (2017), pp. 745–768
• M. Bause, F. A. Radu, U. Köcher, Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space, Numer. Math. 137 (2017), pp. 773–818
• M. Bause, U. Köcher, Variational time discretization for mixed finite element approximations of nonstationary diffusion problems, J. Comput. Appl. Math., 289 (2015), pp. 208–224.
• U. Köcher, M. Bause, Variational space-time methods for the wave equation, J. Sci. Comput., 61 (2014), pp. 424–453
• M. P. Bruchhäuser, K. Schwegler, M. Bause, Dual weighted residual based  error control for nonstationary convection-dominated equations: potential or ballast?, Lecture Notes in Computational Science and Engineering, submitted (2018)
• U. Köcher, M. P. Bruchhäuser, M. Bause, Efficient and scalable data structures and algorithms for goal-oriented adaptivity of space-time FEM codes, submitted, p. 1-6, 2018, arXiv: 1812.08558

The efficient and reliable approximation of nonstationary convection-dominated transport problems continues to remain a challenging task. The solutions of those problems are typically characterized by the occurrence of sharp moving fronts and interior or boundary layers. For their numerical approximation residual based stabilization concepts like the streamline upwind Petrov-Galerkin (SUPG) approach are used. A further and widespread technique to capture singular phenomena and sharp profiles of solutions is the application of adaptive mesh refinement in space and time based on an a posteriori error control mechanism. To design an adaptive method, the availability of an appropriate and robust a posteriori error estimator is required. One possible technique is the commonly used Dual Weighted Residual (DWR) method, where the error is controlled in an arbitrary user-chosen target quantity of physical interest. This offers large potential in obtaining very economical meshes with regard to the underlying goal quantity. Our research focus deals with the combination of both stabilized space-time finite element approximation schemes and adaptive mesh refinement.

For the implementation the in-house parallel frontend software DTM++ is used, which is based on the open source finite element library deal.II. In particular, the group’s methods and results have been published in the following works.

• M. P. Bruchhäuser, U. Köcher, M. Bause, On the implementation of an adaptive multirate framework for coupled transport and flow, J. Sci. Comput. 93, 59 (2022); [https://doi.org/10.1007/s10915-022-02026-z]  [arXiv:2202.03748v1]
• M. Bause, M. P. Bruchhäuser, U. Köcher, Flexible goal-oriented adaptivity for higher-order space-time discretizations of transport problems with coupled flow, Computers and Mathematics with Applications, in press (2020), [https://doi.org/10.1016/j.camwa.2020.08.028] [https://arxiv.org/abs/2002.06855v2]
• U. Köcher, M. P. Bruchhäuser, M. Bause, Efficient and scalable data structures and algorithms for goal-oriented adaptivity of space-time FEM codes, submitted, p. 1-6, 2018, arXiv: 1812.08558
• M. P. Bruchhäuser, K. Schwegler, M. Bause, Dual weighted residual based error control for nonstationary convection-dominated equations: potential or ballast?, Lecture Notes in Computational Science and Engineering, submitted (2018), pp. 1–13; arXiv:1812.06810
• M. P. Bruchhäuser, K. Schwegler, M. Bause, Numerical study of goal- oriented error control for stabilized finite element methods, Lecture Notes in Computational Science and Engineering, submitted (2017), pp. 1–20; arXiv:1803.10643
• K. Schwegler, M. P. Bruchhäuser, M. Bause, Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems, 2018, arXiv: 1601.06544v2.