Mini-Workshop: Adaptive Methods for Control Problems Constrained by Time-Dependent PDEs

Optimization problems constrained by time-dependent PDEs (Partial Differential Equations) are challenging from a computational point of view: even in the simplest case, one needs to solve a system of PDEs coupled globally in time and space for the unknown solutions (the state, the costate and the control of the system). Typical and practically relevant examples are the control of nonlinear heat equations as they appear in laser hardening or the thermic control of flow problems (Boussinesq equations). Specifically for PDEs with a long time horizon, conventional time-stepping methods require an enormous storage of the respective other variables. In contrast, adaptive methods aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities and are, therefore, most promising

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

On the Dependence of the Pressure on the Time Step in Incompressible Flow Simulations on Varying Spatial Meshes

Subject of this paper is an analysis of the behavior of the pressure on dynamically changing spatial meshes during the computation of nonstationary incompressible flows. In particular, we are concerned with discontinuous Galerkin finite element discretizations in time. Here it is observed that whenever the spatial mesh is changed between two time steps the pressure in the next time step will diverge with order $k^{-1}$. We will proof that this behavior is due to the fact, that discrete solenoidal fields lose this property under changes of the spatial discretization. In addition we will numerically study the fractional-step-$theta$ scheme, and discuss why the divergence is not observed when using this time discretization. Finally we will derive a possible way to circumvent this problem

Space-time goal-oriented error control and adaptivity for discretizations and reduced order modeling of multiphysics problems

In this thesis, we investigate the use of adaptive methods for the efficient solution of linear multiphysics problems and nonlinear coupled problems. The main ingredients are a posteriori error estimates based on the dual-weighted residual method. By solving an auxiliary adjoint problem, these error estimates can be used to compute local error indicators for spatial and temporal refinements, which can be used for adaptive spatial and temporal meshes for e.g. the Navier-Stokes equations. For interface- and volume-coupled problems, we present a further extension of temporal adaptivity by using different temporal meshes for each subproblem while still being able to assemble the linear system in a monolithic fashion. Since multiphysics problems, like poroelasticity, are expensive to solve for fine discretizations with millions of degrees of freedom, we present a novel online-adaptive model order reduction method called MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates), which merges classical proper orthogonal decomposition based model order reduction with a posteriori error estimates. Thus, we can avoid the costly offline phase of classical model order reduction methods and still achieve high accuracy by enriching the reduced basis on-the-fly in the online phase when the error estimators exceed a given tolerance.German Research Foundation (DFG)/International Research Training Group 2657/Grant Number 43308229/E

A dynamically adaptive mesh method for internal flows

A dynamic solution adaptive mesh method was implemented into a finite-volume numerical method for solving unsteady flowfields described by the two-dimensional, unsteady, Navier-Stokes and Euler equations. The objective was to improve the resolution and accuracy of solutions which contained flow gradients which varied in strength and position with time. Variational principles were used to formulate the mesh equations with which meshes were generated to have the desired smoothness, orthogonality, and volume adaption qualities. The adaption of the mesh to the flow solution was driven by the presence of flow gradients. The dynamics of the mesh was accounted for in the flow equations through the mesh speeds. A comparison was made between one approach which computed the mesh speeds from a backwards time differences of the mesh and another approach which computed the mesh speeds from a system of mesh speed equations which were derived from the time differentiation of the mesh equations. The dynamically adaptive mesh method was demonstrated for model problems involving solution and boundary dynamics, inviscid flows in a converging-diverging nozzle, viscous boundary-layer flows over flat plates, and viscous flows in a transonic diffuser. It was found that the approach using the mesh speed equations was more accurate than the approach using the time-differenced mesh speeds. There was difficulty is obtaining proper clustering of the meshes for viscous flows

Adaptive high-resolution finite element schemes

The numerical treatment of flow problems by the finite element method is addressed. An algebraic approach to constructing high-resolution schemes for scalar conservation laws as well as for the compressible Euler equations is pursued. Starting from the standard Galerkin approximation, a diffusive low-order discretization is constructed by performing conservative matrix manipulations. Flux limiting is employed to compute the admissible amount of compensating antidiffusion which is applied in regions, where the solution is sufficiently smooth, to recover the accuracy of the Galerkin finite element scheme to the largest extent without generating non-physical oscillations in the vicinity of steep gradients. A discrete Newton algorithm is proposed for the solution of nonlinear systems of equations and it is compared to the standard fixed-point defect correction approach. The Jacobian operator is approximated by divided differences and an edge-based procedure for matrix assembly is devised exploiting the special structure of the underlying algebraic flux correction (AFC) scheme. Furthermore, a hierarchical mesh adaptation algorithm is designed for the simulation of steady-state and transient flow problems alike. Recovery-based error indicators are used to control local mesh refinement based on the red-green strategy for element subdivision. A vertex locking algorithm is developed which leads to an economical re-coarsening of patches of subdivided cells. Efficient data structures and implementation details are discussed. Numerical examples for scalar conservation laws and the compressible Euler equations in two dimensions are presented to assess the performance of the solution procedure.In dieser Arbeit wird die numerische Simulation von skalaren Erhaltungsgleichungen sowie von kompressiblen Eulergleichungen mit Hilfe der Finite-Elemente Methode behandelt. Dazu werden hochauflösende Diskretisierungsverfahren eingesetzt, welche auf algebraischen Konstruktionsprinzipien basieren. Ausgehend von der Galerkin-Approximation wird eine Methode niedriger Ordnung konstruiert, indem konservative Matrixmodifikationen durchgeführt werden. Anschließend kommt ein sog. Flux-Limiter zum Einsatz, der in Abhängigkeit von der lokalen Glattheit der Lösung den zulässigen Anteil an Antidiffusion bestimmt, die zur Lösung der Methode niedriger Ordnung hinzuaddiert werden kann, ohne dass unphysikalische Oszillationen in der Nähe von steilen Gradienten entstehen. Die resultierenden nichtlinearen Gleichungssysteme können entweder mit Hilfe von Fixpunkt-Defektkorrektur-Techniken oder mittels diskreter Newton-Verfahren gelöst werden. Für letztere wird die Jacobi-Matrix mit dividierten Differenzen approximiert, wobei ein effizienter, kantenbasierter Matrixaufbau aufgrund der speziellen Struktur der zugrunde liegenden Diskretisierung möglich ist. Ferner wird ein hierarchischer Gitteradaptionsalgorithmus vorgestellt, welcher sowohl für die Simulation von stationären als auch zeitabhängigen Strömungen geeignet ist. Die lokale Gitterverfeinerung folgt dem bekannten Rot-Grün Prinzip, wobei rekonstruktionsbasierte Fehlerindikatoren zur Markierung von Elementen zum Einsatz kommen. Ferner erlaubt das sukzessive Sperren von Knoten, die nicht gelöscht werden können, eine kostengünstige Rückvergröberung von zuvor unterteilten Elementen. In der Arbeit wird auf verschiedene Aspekte der Implementierung sowie auf die Wahl von effizienten Datenstrukturen zur Verwaltung der Gitterinformationen eingegangen. Der Nutzen der vorgestellten Simulationswerkzeuge wird anhand von zweidimensionalen Beispielrechnungen für skalare Erhaltungsgleichungen sowie für die kompressiblen Eulergleichungen analysiert