Uwe Köcher

Markus Bause, Florin A. Radu, Uwe Köcher

Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods (MFEM) in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin-Petrov time discretization schemes that is combined with a mixed finite element (MFE) approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach-Nečas-Babuška theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.

Markus Bause, Uwe Köcher, Florin A. Radu et al.

We introduce and analyze a post-processing for a family of variational space-time approximations to wave problems. The discretization in space and time is based on continuous finite element methods. The post-processing lifts the fully discrete approximations in time from continuous to continuously differentiable ones. Further, it increases the order of convergence of the discretization in time which can be be exploited nicely, for instance, for a-posteriori error control. The convergence behavior is shown by proving error estimates of optimal order in various norms. A bound of superconvergence at the discrete times nodes is included. To show the error estimates, a special approach is developed. Firstly, error estimates for the time derivative of the post-processed solution are proved. Then, in a second step these results are used to establish the desired error estimates for the post-processed solution itself. The need for this approach comes through the structure of the wave equation providing only stability estimates that preclude us from using absorption arguments for the control of certain error quantities. A further key ingredient of this work is the construction of a new time-interpolate of the exact solution that is needed in an essential way for deriving the error estimates. Finally, a conservation of energy property is shown for the post-processed solution which is a key feature for approximation schemes to wave equations. The error estimates given in this work are confirmed by numerical experiments.

Jakub Wiktor Both, Uwe Köcher

We study the numerical solution of the quasi-static linear Biot's equations solved iteratively by the fixed-stress splitting scheme. In each iteration the mechanical and flow problems are decoupled, where the flow problem is solved by keeping an artificial mean stress fixed. This introduces a numerical tuning parameter which can be optimized. We investigate numerically the optimality of the parameter and compare our results with physically and mathematically motivated values from the literature, which commonly only depend on mechanical material parameters. We demonstrate, that the optimal value of the tuning parameter is also affected by the boundary conditions and material parameters associated to the fluid flow problem suggesting the need for the integration of those in further mathematical analyses optimizing the tuning parameter.

Uwe Köcher, Markus Bause

We study higher-order space-time variational discretisations for modeling complex processes in porous media that include fluid and structure interactions which are of fundamental importance in many engineering fields with applications in subsurface processes, battery-design and biomechanics. For the discretisation in time we deploy discontinuous Galerkin dG(r) and continuous Galerkin cG(q) discretisation families. Moreover we introduce a new coupled dG(r)-cG(q) mixed time discretisation and show numerically the stability advantages in the case of incompatible initial data under massively reduced computational costs. For the discretisation in space we use a mixed finite element method for the flow problem to ensure local mass conservation and a continuous Galerkin method for the mechanics. We consider solving sequentially the coupling of flow and mechanics with the fixed-stress iterative approach such that we can reuse our system solver and preconditioning technologies for the arising block system matrices from higher-order in time discretisations. Numerical experiments show firstly the undeniable advantages of discontinuous Galerkin time discretisations dG(0) and dG(1) over the continuous Galerkin time discretisation cG(1) in the case of incompatible initial data, secondly the advantages of the new coupled dG(1)-cG(1) in time approach in the case of incompatible initial data with massively reduced computational costs and better accuracy compared to the dG(1) time discretisation and thirdly the performance and efficiency differences of the dG(0), cG(1), dG(1) and the new dG(1)-cG(1) fixed-stress solver approaches for a sophisticated and physically relevant three-dimensional numerical example.

Uwe Köcher

The accurate, reliable and efficient numerical approximation of multi-physics processes in heterogeneous porous media with varying media coefficients that include fluid flow and structure interactions is of fundamental importance in energy, environmental, petroleum and biomedical engineering applications fields for instance. Important applications include subsurface compaction drive, carbon sequestration, hydraulic and thermal fracturing and oil recovery. Biomedical applications include the simulation of vibration therapy for osteoporosis processes of trabeculae bones, estimating stress levels induced by tumour growth within the brain or next-generation spinal disc prostheses. Variational space-time methods offers some appreciable advantages such as the flexibility of the triangulation for complex geometries in space and natural local time stepping, the straightforward construction of higher-order approximations and the application of efficient goal-oriented (duality-based) adaptivity concepts. In addition to that, uniform space-time variational methods appear to be advantageous for stability and a priori error analyses of the discrete schemes. Especially (high-order) discontinuous in time approaches appear to have favourable properties due to the weak application of the initial conditions. The development of monolithic multi-physics schemes, instead of iterative coupling methods between the physical problems, is a key component of the research to reduce the modeling error. Special emphasis is on the development of efficient multi-physics and multigrid preconditioning technologies and their implementation. The simulation software DTM++ is a modularised framework written in C++11 and builds on top of deal.II toolchains. The implementation allows parallel simulations from notebooks up to cluster scale.

Uwe Köcher

Interior penalty discontinuous Galerkin discretisations (IPDG) and especially the symmetric variant (SIPG) for time-domain wave propagation problems are broadly accepted and widely used due to their advantageous properties. Linear systems with block structure arise by applying space-time discretisations and reducing the global system to time-slab problems. The design of efficient and robust iterative solvers for linear systems from interior penalty discretisations for hyperbolic wave equations is still a challenging task and relies on understanding the properties of the systems. In this work the numerical properties such as the condition number and the distribution of eigenvalues of different representations of the linear systems coming from space-time discretisations for elastic wave propagation are numerically studied. These properties for interior penalty discretisations depend on the penalisation and on the time interval length.