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In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier-Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor-Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition of unity based error localization. We derive the space-time Newton method for the Navier-Stokes equations and substantiate our methodology on 2D benchmark problems from computational fluid mechanics., Comment: 34 pages, 13 figures

In this work, a multirate in time approach resolving the different time scales of a convection-dominated transport and coupled fluid flow is developed and studied in view of goal-oriented error control by means of the Dual Weighted Residual (DWR) method. Key ingredients are an arbitrary degree discontinuous Galerkin time discretization of the underlying subproblems, an a posteriori error representation for the transport problem coupled with flow and its implementation using space-time tensor-product spaces. The error representation allows the separation of the temporal and spatial discretization error which serve as local error indicators for adaptive mesh refinement. The performance of the approach and its software implementation are studied by numerical convergence examples as well as an example of physical interest for convection-dominated transport., Comment: arXiv admin note: text overlap with arXiv:2002.06855

We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modeling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in time and inf-sup stable pairs of finite element spaces for the spatial variables are investigated. Optimal order error estimates are proved by an analysis in weighted norms that depict the energy of the system's unknowns. A further important ingredient and challenge of the analysis is the control of the couplings terms. The techniques developed here can be generalized to other families of Galerkin space discretizations and advanced models. The error estimates are confirmed by numerical experiments, also for higher order piecewise polynomials in time and space. The latter lead to algebraic systems with complex block structure and put a facet of challenge on the design of iterative solvers. An efficient solution technique is referenced.

In this work, a flexible higher-order space-time adaptive finite element approximation of convection-dominated transport with coupled fluid flow is developed and studied. Convection-dominated transport is a challenging subproblem in poromechanics in which coupled transport with flow, chemical reaction and mechanical response in porous media is considered. Key ingredients are the arbitrary degree discontinuous Galerkin time discretization of the primal and dual problems for the Dual Weighted Residual (DWR) approach, an a posteriori error estimation for the transport problem coupled with flow and its implementation in an advanced software architecture. The error estimate allows the separation of the temporal and spatial discretization error contributions which facilitates the simultaneous adjustment of the time and space mesh. The performance of the approach and its software implementation is studied by numerical convergence examples as well as an example of physical interest for convection-dominated cases.

The cost- and memory-efficient numerical simulation of coupled volume-based multi-physics problems like flow, transport, wave propagation and others remains a challenging task with finite element method (FEM) approaches. Goal-oriented space and time adaptive methods derived from the dual weighted residual (DWR) method appear to be a shiny key technology to generate optimal space-time meshes to minimise costs. Current implementations for challenging problems of numerical screening tools including the DWR technology broadly suffer in their extensibility to other problems, in high memory consumption or in missing system solver technologies. This work contributes to the efficient embedding of DWR space-time adaptive methods into numerical screening tools for challenging problems of physically relevance with a new approach of flexible data structures and algorithms on them, a modularised and complete implementation as well as illustrative examples to show the performance and efficiency.

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.

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.