arXiv subjects

Markus Bause

Publications and source records attributed to Markus Bause.

At least 37 records · Page 2Linked to original sources

arXiv2022

A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport

In this work, a cost-efficient space-time adaptive algorithm based on the Dual Weighted Residual (DWR) method is developed and studied for a coupled model problem of flow and convection-dominated transport. Key ingredients are a multirate approach adapted to varying dynamics in time of the subproblems, weighted and non-weighted error indicators for the transport and flow problem, respectively, and the concept of space-time slabs based on tensor product spaces for the data structure. In numerical examples the performance of the underlying algorithm is studied for benchmark problems and applications of practical interest. Moreover, the interaction of stabilization and goal-oriented adaptivity is investigated for strongly convection-dominated transport.

math.NA↗

arXiv2022

Efficiency of local Vanka smoother geometric multigrid preconditioning for space-time finite element methods to the Navier-Stokes equations

Numerical simulation of incompressible viscous flow, in particular in three space dimensions, continues to remain a challenging task. Space-time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Linearizing the resulting algebraic problems by Newton's method yields linear systems with block matrices built of $(k+1)\times (k+1)$ saddle point systems, where $k$ denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a $V$-cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two- and three-dimensional benchmark problem of flow around a cylinder. Here, the robustness of the solver with respect to the piecewise polynomial order $k$ in time is analyzed and proved numerically.

math.NA↗

arXiv2022

A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier--Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for GMRES iterations. Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the \emph{deal.II} finite element library. The concepts are flexible and can be transferred to similar software platforms.

math.NA↗

arXiv2022

On the implementation of an adaptive multirate framework for coupled transport and flow

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.

math.NA↗

arXiv2021

CutFEM and ghost stabilization techniques for higher order space-time discretizations of the Navier-Stokes equations

We propose and analyze computationally 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 potential of such cut finite element techniques for higher order space-time finite element methods has rarely been studied in the literature so far and deserves further elucidation. 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. Thereby, an expensive remeshing and adaptation of the sparse matrix data structure are avoided and the computations are accelerated. To prevent spurious oscillations caused by irregular intersections of mesh cells, a penalization, defining also implicitly the extension to ghost 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 parallel implementation of the matrix assembly is described. The optimal order convergence properties of the algorithm are illustrated in a numerical experiment for an evolving domain. The well-known 2d benchmark of flow around a cylinder as well as flow around moving obstacles with arising cut cells and fictitious domains are considered further.

math.NA↗

arXiv2021

$C^1$-conforming variational discretization of the biharmonic wave equation

Biharmonic wave equations are of importance to various applications including thin plate analyses. In this work, the numerical approximation of their solutions by a $C^1$-conforming in space and time finite element approach is proposed and analyzed. Therein, the smoothness properties of solutions to the continuous evolution problem is embodied. High potential of the presented approach for more sophisticated multi-physics and multi-scale systems is expected. Time discretization is based on a combined Galerkin and collocation technique. For space discretization the Bogner--Fox--Schmit element is applied. Optimal order error estimates are proven. The convergence and performance properties are illustrated with numerical experiments.

math.NA↗

arXiv2021

Numerical convergence of discrete extensions in a space-time finite element, fictitious domain method for the Navier-Stokes equations

A key ingredient of our fictitious domain, higher order space-time cut finite element (CutFEM) approach for solving the incompressible Navier--Stokes equations on evolving domains (cf.\ \cite{Bause2021}) is the extension of the physical solution from the time-dependent flow domain $Ω_f^t$ to the entire, time-independent computational domain $Ω$. The extension is defined implicitly and, simultaneously, aims at stabilizing the discrete solution in the case of unavoidable irregular small cuts. Here, the convergence properties of the scheme are studied numerically for variations of the combined extension and stabilization.

math.NA↗

arXiv2021

Prediction of Ultrasonic Guided Wave Propagation in Solid-fluid and their Interface under Uncertainty using Machine Learning

Structural health monitoring (SHM) systems use the non-destructive testing principle for damage identification. As part of SHM, the propagation of ultrasonic guided waves (UGWs) is tracked and analyzed for the changes in the associated wave pattern. These changes help identify the location of a structural damage, if any. We advance existing research by accounting for uncertainty in the material and geometric properties of a structure. The physics model used in this study comprises of a monolithically coupled system of acoustic and elastic wave equations, known as the wave propagation in fluid-solid and their interface (WpFSI) problem. As the UGWs propagate in the solid, fluid, and their interface, the wave signal displacement measurements are contrasted against the benchmark pattern. For the numerical solution, we develop an efficient algorithm that successfully addresses the inherent complexity of solving the multiphysics problem under uncertainty. We present a procedure that uses Gaussian process regression and convolutional neural network for predicting the UGW propagation in a solid-fluid and their interface under uncertainty. First, a set of training images for different realizations of the uncertain parameters of the inclusion inside the structure is generated using a monolithically-coupled system of acoustic and elastic wave equations. Next, Gaussian processes trained with these images are used for predicting the propagated wave with convolutional neural networks for further enhancement to produce high-quality images of the wave patterns for new realizations of the uncertainty. The results indicate that the proposed approach provides an accurate prediction for the WpFSI problem in the presence of uncertainty.

eess.SP↗

arXiv2020

Higher order Galerkin-collocation time discretization with Nitsche's method for the Navier-Stokes equations

We propose and study numerically the implicit approximation in time of the Navier-Stokes equations by a Galerkin-collocation method in time combined with inf-sup stable finite element methods in space. The conceptual basis of the Galerkin-collocation approach is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs in terms of less complex algebraic systems of the latter. Regularity of higher order in time of the discrete solution is ensured further. As an additional ingredient, we employ Nitsche's method to impose all boundary conditions in weak form with the perspective that evolving domains become feasible in the future. We carefully compare the performance poroperties of the Galerkin-collocation approach with a standard continuous Galerkin-Petrov method using piecewise linear polynomials in time, that is algebraically equivalent to the popular Crank-Nicholson scheme. The condition number of the arising linear systems after Newton linearization as well as the reliable approximation of the drag and lift coefficient for laminar flow around a cylinder (DFG flow benchmark with $Re=100$) are investigated. The superiority of the Galerkin-collocation approach over the linear in time, continuous Galerkin-Petrov method is demonstrated therein.

math.NA↗

arXiv2020

Flexible goal-oriented adaptivity for higher-order space-time discretizations of transport problems with coupled flow

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.

math.NA↗

arXiv2019

Iterative Coupling for Fully Dynamic Poroelasticity

We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge-Kutta methods. Recasting the semi-discrete solution as the minimizer of a properly defined energy functional, the proof of convergence uses its alternating minimization. The scheme is closely related to the undrained split for the quasi-static Biot system.

math.NA↗

arXiv2019

Galerkin-collocation approximation in time for the wave equation and its post-processing

We introduce and analyze a class of Galerkin-collocation discretization schemes in time for the wave equation. Its 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. Continuously differentiable in time discrete solutions are obtained by the application of a special quadrature rule involving derivatives. Optimal order error estimates are proved for fully discrete approximations based on the Galerkin-collocation approach. Further, the concept of Galerkin-collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct connection between the two families by a computationally cheap post-processing is presented. The error estimates are illustrated by numerical experiments.

math.NA↗

arXiv2019

Numerical study of Galerkin-collocation approximation in time for the wave equation

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.

math.NA↗

arXiv2018

Efficient and scalable data structures and algorithms for goal-oriented adaptivity of space-time FEM codes

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.

cs.MS↗

arXiv2018

Dual weighted residual based error control for nonstationary convection-dominated equations: potential or ballast?

Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.

math.NA↗

arXiv2018

A mixed discontinuous-continuous Galerkin time discretisation for Biot's system

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.

math.NA↗

arXiv2018

Numerical study of goal-oriented error control for stabilized finite element methods

The efficient and reliable approximation of convection-dominated problems continues to remain a challenging task. To overcome the difficulties associated with the discretization of convection-dominated equations, stabilization techniques and a posteriori error control mechanisms with mesh adaptivity were developed and studied in the past. Nevertheless, the derivation of robust a posteriori error estimates for standard quantities and in computable norms is still an unresolved problem and under investigation. Here we combine the Dual Weighted Residual (DWR) method for goal-oriented error control with stabilized finite element methods. By a duality argument an error representation is derived on that an adaptive strategy is built. The key ingredient of this work is the application of a higher order discretization of the dual problem in order to make a robust error control for user-chosen quantities of interest feasible. By numerical experiments in 2D and 3D we illustrate that this interpretation of the DWR methodology is capable to resolve layers and sharp fronts with high accuracy and to further reduce spurious oscillations.

math.NA↗

arXiv2018

Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems

The numerical approximation of convection-dominated problems continues to remain subject of strong interest. Families of stabilization techniques for finite element methods were developed in the past. Adaptive techniques based on a posteriori error estimates offer potential for further improvements. However, there is still a lack in robust a posteriori error estimates in natural norms of the discretizations. Here we combine the dual weighted residual method for goal-oriented error control with stabilized finite element approximations. By a duality argument an error representation is derived on that a space-time adaptive approach is built. It differs from former works on the dual weighted residual method. Numerical experiments illustrate that our schemes are capable to resolve layers and sharp fronts with high accuracy and to further reduce spurious oscillations of approximations.

math.NA↗