## 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

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- Finite-Elemente-Methode (4)
- Anisotropie (1)
- Elektromagnetisches Feld (1)
- Elektrostatik (1)
- FEM Software (1)
- Fehlerabschätzung (1)
- Finite-Volumen-Methode (1)
- Galerkin-Methode (1)
- Gekoppeltes System (1)
- Magnetostatik (1)

The simulation of large-scale coupled field problems with finite elements is a common task nowadays. However, the problem size is often limited by the presence of different length scales, either due to the geometric setup or the different field characteristics of the fields quantities involved, like boundary layers or different wavelengths. For interface-coupled problems, non-conforming-techniques like the Mortar or Nitsche method can be applied, while for single-field and volumetric problems dimensionally-reduced approximations like shells and plates are preferred. However, the latter ones have limitations due to modeling errors, missing ability to represent structural waves, deteriorated numerical behavior, as well as limitations when coupling to volumetric elements.
Thus, the subject of this thesis is the development of a pure volumetric discretization approach for coupled and anisotropic field problems by means of finite elements of higher order, also known as the p-version finite element method (FEM). The anisotropic field characteristics in case of thin-structured and multi-field problems is taken into account by a field-wise and directional dependent choice of the polynomial degree p, which extends the original ideas developed by Szabo and Düster. Additional attention is given to the efficient numerical solution of the arising algebraic systems for the case of low-frequency and static magnetic problems. For all physical fields, special emphasis is put on the correct functional space within the framework of higher order elements.
For thin piezoelectric structures, detailed investigations based on the classical plate theory are revisited to derive guidelines for the approximation of the electric field. An extensive parameter study provides generalizations for the suitable polynomial degree for plate-like mechanical structures. The guidelines are investigated for the low-frequency case, with extensions to wave propagation and eigenfrequency analysis. The applicability is shown by means of practical actuator and sensor problems.
The adaption to thin magnetic structures with highly permeable parts is performed by using anisotropic polynomial degrees. The numerical problems arising from the discretization of thin-structured problems are solved by means of a novel anisotropic p-version preconditioner with no a-priori knowledge about the thin directions and an improved static condensation approach. For nonlinear problems, the calculation can be accelerated further by utilizing a two-level solution technique, being natural to the p-version approach of nested polynomial spaces.
Finally, for transient coupled magneto-mechanical problems based on Lorentz forces, a robust skin-depth resolving method based on a boundary layer analysis is proposed. Its applicability is demonstrated by means of a simplified simulation model of a clinical magnetic resonance imaging (MRI) scanner.

In this work several time discretization techniques for capillary flows, i.e. either one-phase free
surface flow or two-phase flow, are compared. The focus is the development of methods that
are a) of higher order (i.e. at least second order), b) unconditionally stable and c) do not suffer
from much numerical dissipativity.
The mathematical model of the problem is initially described and a variational, dimensionless
formulation is given that can be used for both one- and two-phase capillary flows. The formula-
tion is in Arbitrary Lagrangian-Eulerian (ALE) coordinates so a problem-adapted moving mesh
can be used. Utilizing the Finite Element Method (FEM) the equations are discretized in space
and a differential algebraic matrix-vector formulation is derived.
Several fully implicit and linearly implicit time discretization techniques are introduced, their
properties with respect to stability, convergence, and dissipativity are discussed. These methods
are then applied to discretize the equations in time. Thus fully discrete equations are derived
that can be used for computer simulations.
Furthermore a space-time Galerkin approach is presented, its stability, convergence and
dissipativity properties are discussed as well. This method is then applied to the variational
formulation of the capillary flow problems and a fully discrete system is derived. For the
space-time approach an energy estimate is proved that establishes the unconditional stability
of the method by analytical means.
In an application chapter several one- and two-phase exemplary problems are introduced
and solved with the described time discretization techniques. Using Finite Element simulations
the convergence and dissipativity properties are numerically investigated.
A concluding comparison of the methods presents the advantages and drawbacks of all
methods that were compared.

This dissertation is concerned with the numerical approximation of the incompressible Navier-Stokes equation. As our main contribution, we propose and analyze a new stabilized mixed finite element scheme for computing incompressible laminar flows, and investigate several properties of this new scheme. The finite element analysis is presented for the Oseen model problem. The scheme is based on the Scott-Vogelius mixed finite element and on symmetric stabilization operators for dominant convection, proposed in the last recent years. In particular, the scheme delivers divergence-free pointwise velocity approximations, with an approximation quality that is completely independent of the pressure. Therefore, the cumbersome grad-div stabilization drops out from the discrete equations, and standard multi-grid approaches are possible for the solution of the evolving linear systems. We propose such a multi-grid method and deliver a multi-grid analysis for the full symmetric part of the discrete Oseen equation, including the symmetric stabilization operator for dominant convection. We further propose a coupled FEM-FVM scheme for convection-diffusion problems in an incompressible flow field. For the numerical computation of a stationary incompressible Navier-Stokes equation we propose the above stabilized Scott-Vogelius scheme, and for a stationary or nonstationary scalar convection-diffusion equation, we propose a Voronoi-box-based finite volume scheme on boundary conforming Delaunay meshes. Since Scott-Vogelius finite element approximations are divergence-free pointwise, we can prove a discrete maximum principle for the discrete convection-diffusion equation. Finally, we present several numerical examples, in order to illustrate in which situations the proposed stabilized Scott-Vogelius scheme delivers more accurate numerical approximations than other approaches. Here, a 2D example of a colliding flow might deserve further interest, since it seems to have some physical relevance. Further, we present a 3D application from electrochemistry, where the coupled FEM-FVM scheme seems to be promising due to the establishment of a discrete maximum principle.

The subject of this thesis is an adaptive finite element method for the different kinds of partial differential equations. This topic is extremely popular nowadays since an accurate computation of the real-life engineering problems requires large number of unknowns. That is very costly with respect to computer time. Therefore, developing numerical methods which are more efficient is a desirable goal. Adaptive finite element methods in which a mesh (discretization of the computational domain) changes to improve a solution of the problem, can significantly reduce a demand for large number of unknowns. Changing of mesh is controlled and governed by an {\it a posteriori} error estimation of the solution. At first, a post-processing error estimation and adaptive procedures based on it are studied. {\it A posteriori} local and global error estimates based on a patch recovery technique are derived. A method for computation of an error distribution over the computational mesh and a remeshing strategy for construction of an optimal mesh based on the error distribution are presented. An adaptive automatic mesh generation procedure is discussed. It adjusts automatically the initial coarse computational mesh so, that the estimated errors are controlled within a specified tolerance. Algorithms for obtaining a regular refined grid, which means mesh without hanging nodes, in two and three dimensional spaces for different types of mesh are described. The presented adaptive procedures are tested on electrostatic problems in two and three dimensional spaces. The results are compared to non-adaptive numerical simulations. Then, adaptive procedures for a harmonic analysis of the acoustic wave equation are developed. Procedures for a local and global error estimation, a recovery and a remeshing technique are introduced. Numerical examples in two-dimensional space confirm the reliability of the developed adaptive finite element method for a frequency domain analysis of an acoustic equation. Next, a recovery technique for N\'ed\'elec's edge finite elements is developed. Local and global error indicators, which are based on energy errors for electromagnetic problems with discontinuity of magnetic potential across an interface of two different medium are introduced. The efficiency of the developed adaptive finite element method performance is proved on an example in 3D case. Finally, adaptive numerical simulations for industrial applications such as computation of the electric field of an voltage-driven bar, standing waves in a model of cleaning bath and a magnetic flux of problem TEAM 20 are conducted. By means of the adaptive finite element, a sufficiently accurate solution for these problems are received.