Markus Bause

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.

Marius Paul Bruchhäuser, Kristina Schwegler, Markus Bause

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.

Marius Paul Bruchhäuser, Kristina Schwegler, Markus Bause

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.

Markus Bause

We analyze an iterative coupling of mixed and discontinuous Galerkin methods for numerical modelling of coupled flow and mechanical deformation in porous media. The iteration is based on an optimized fixed-stress split along with a discontinuous variational time discretization. For the spatial discretization of the subproblem of flow mixed finite element techniques are applied. The discretization of the subproblem of mechanical deformation uses discontinuous Galerkin methods. They have shown their ability to eliminate locking that sometimes arises in numerical algorithms for poroelasticity and causes nonphysical pressure oscillations.

Kristina Schwegler, Marius P. Bruchhäuser, Markus Bause

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.

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.

Moataz Dawor, Nils Margenberg, Markus Bause

Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. Due to the high dimensionality, the solution of the arising algebraic systems do not become feasible without efficient solvers, preconditioners, and software architectures. A stochastic Galerkin discretization with an embedded slabwise finite element approximation of the space and time variables is proposed and analyzed numerically. For solving the linear systems, GMRES iterations are block-preconditioned by a geometric multigrid (GMG) technique using a local Vanka smoother for the space-time subsystems. Monte-Carlo methods are also used for solving random parabolic problems and studied here for the purpose of comparison. The Monte-Carlo approach is built on the space-time finite element formulation together with the GMRES-GMG solver technology. All algorithms have been implemented in a unified matrix-free framework based on the deal.II software library. Comparative numerical evaluations illustrate the performance properties of both approaches, including convergence of the discretizations and statistics of the algebraic solver. Superiority of the stochastic Galerkin approach is observed.

Subhayan De, Bhuiyan Shameem Mahmood Ebna Hai, Alireza Doostan et al.

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.