Schnelle Löser für Partielle Differentialgleichungen

This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds

Proceedings for the ICASE Workshop on Heterogeneous Boundary Conditions

Domain Decomposition is a complex problem with many interesting aspects. The choice of decomposition can be made based on many different criteria, and the choice of interface of internal boundary conditions are numerous. The various regions under study may have different dynamical balances, indicating that different physical processes are dominating the flow in these regions. This conference was called in recognition of the need to more clearly define the nature of these complex problems. This proceedings is a collection of the presentations and the discussion groups

Fast iterative solvers for Cahn-Hilliard problems

Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, Dissertation, 2016von M. Sc. Jessica BoschLiteraturverzeichnis: Seite [247]-25

Multi space reduced basis preconditioners for parametrized partial differential equations

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs. In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation. The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system. We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency. We apply our strategy to a broad range of problems described by parametrized PDEs: (i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations. Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results. Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques. Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling

Boundary layer instabilities due to surface irregularities: a harmonic Navier-Stokes approach

Maintaining laminar flow and delaying transition to turbulence on aircraft wings reduces friction drag and hence fuel consumption for an improved ecological footprint. Nonetheless, widespread models of disturbance growth in boundary layers discard important transition stages and are inadequate to incorporate the effect of surface irregularities causing rapid variations in the underlying steady flow. This thesis applies global or Harmonic Navier-Stokes (HNS) methods to quantify the growth of instabilities in shear flows with two inhomogeneous spatial directions. Such methods deliver greater fidelity than the standard Parabolised Stability Equations (PSE). This work presents an efficient parallel computational framework to solve linear and non-linear HNS problems. We use BiGlobal analysis to investigate the existence of temporally unstable modes on a flat plate with smooth indentations featuring laminar separation bubbles (LSBs). Then, for the first time, it is applied to a swept-wing boundary layer featuring Backward- and Forward-Facing Steps (BFSs and FFSs). Localised unstable modes are identified for step heights exceeding the local boundary-layer displacement thickness of the clean geometry. BFSs are found to be more destabilising than equivalent FFSs, especially in the presence of the LSB formed behind the infinite-swept BFS. Next, we introduce the non-linear HNS method as an improvement over the non-linear PSE, able to model receptivity and non-linear mode interaction at a fraction of the cost of Direct Numerical Simulation. The method can model flow destabilisation scenarios on swept wings exhibiting surface features and holds the potential for accurate transition prediction. Its performance is assessed in the case of a Tollmien-Schlichting wave interacting with a cylindrical roughness located on a nearly flat aerofoil section. Finally, we consider crossflow disturbances generated by placing Discrete Roughness Elements (DRE) at the leading edge of a swept wing and follow their non-linear development up to a strongly saturated state. Non-linear receptivity effects are found to arise with increasing DRE heights.Open Acces

Learned infinite elements for helioseismology

This thesis presents efficient techniques for integrating the information contained in the Dirichlet-to-Neumann (DtN) map of time-harmonic waves propagating in a stratified medium into finite element discretizations. This task arises in the context of domain decomposition methods, e.g. when reducing a problem posed on an unbounded domain to a bounded computational domain on which the problem can then be discretized. Our focus is on stratified media like the Sun, that allow for strong reflection of waves and for which suitable methods are lacking. We present learned infinite elements as a possible approach to deal with such media utilizing the assumption of a separable geometry. In this case, the DtN map is separable, however, it remains a non-local operator with a dense matrix representation, which renders its direct use computationally inefficient. Therefore, we approximate the DtN only indirectly by adding additional degrees of freedom to the linear system in such a way that the Schur complement w.r.t. the latter provides an optimal approximation of DtN and sparsity of the linear system is preserved. This optimality is ensured via the solution of a small minimization problem, which incorporates solutions of one-dimensional time-harmonic wave equations and allows for great flexibility w.r.t. properties of the medium. In the first half of the thesis we provide an error analysis of the proposed method in a generic framework which demonstrates that exponentially fast convergence rates can be expected. Numerical experiments for the Helmholtz equation and an in-depth study on modelling the solar atmosphere with learned infinite elements demonstrate the high accuracy and flexibility of the proposed method in practical applications. In the second half of the thesis, the potential of learned infinite elements in the context of sweeping preconditioners for the efficient iterative solution of large linear systems is investigated. Even though learned infinite elements are very suitable for separable media, they can only be used for tiny perturbations thereof since the corresponding DtN maps turn out to be extremely sensitive to perturbations in the presence of strong reflections.2021-12-2