A priori and a posteriori error estimates for the quad-curl eigenvalue problem

In this paper, we consider a priori and a posteriori error estimates of the H(curl2)-conforming finite element when solving the quad-curl eigenvalue problem. An a priori estimate of eigenvalues with convergence order 2(s − 1) is obtained if the corresponding eigenvector u ∈ Hs − 1(Ω) and ∇ × u ∈ Hs(Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the errors of eigenvectors in the energy norm and upper bounds for the errors of eigenvalues. Numerical examples are presented for validation

A parallel multigrid method for band structure computation of 3D photonic crystals with higher order finite elements

The band structure computation turns into solving a family of Maxwell eigenvalue problems on the periodicity domain. The discretization is done by the finite element method with special higher order H(curl)- and H1-conforming modified elements. The eigenvalue problem is solved by a preconditioned iterative eigenvalue solver with a projection onto the divergence-free vector fields. As a preconditioner we use the parallel multigrid method with a special Hiptmair smoother

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Scalable parallel simulation of variably saturated flow

In this thesis we develop highly accurate simulation tools for variably saturated flow through porous media able to take advantage of the latest supercomputing resources. Hence, we aim for parallel scalability to very large compute resources of over 105 CPU cores. Our starting point is the parallel subsurface flow simulator ParFlow. This library is of widespread use in the hydrology community and known to have excellent parallel scalability up to 16k processes. We first investigate the numerical tools this library implements in order to perform the simulations it was designed for. ParFlow solves the governing equation for subsurface flow with a cell centered finite difference (FD) method. The code targets high performance computing (HPC) systems by means of distributed memory parallelism. We propose to reorganize ParFlow's mesh subsystem by using fast partitioning algorithms provided by the parallel adaptive mesh refinement (AMR) library p4est. We realize this in a minimally invasive manner by modifying selected parts of the code to reinterpret the existing mesh data structures. Furthermore, we evaluate the scaling performance of the modified version of ParFlow, demonstrating excellent weak and strong scaling up to 458k cores of the Juqueen supercomputer at the Jülich Supercomputing Centre. The above mentioned results were obtained for uniform meshes and hence without explicitly exploiting the AMR capabilities of the p4est library. A natural extension of our work is to activate such functionality and make ParFlow a true AMR application. Enabling ParFlow to use AMR is challenging for several reasons: It may be based on assumptions on the parallel partition that cannot be maintained with AMR, it may use mesh-related metadata that is replicated on all CPUs, and it may assume uniform meshes in the construction of mathematical operators. Additionally, the use of locally refined meshes will certainly change the spectral properties of these operators. In this work, we develop an algorithmic approach to activate the usage of locally refined grids in ParFlow. AMR allows meshes where elements of different size neighbor each other. In this case, ParFlow may incur erroneous results when it attempts to communicate data between inter-element boundaries. We propose and discuss two solutions to this issue operating at two different levels: The first manipulates the indices of the degrees of freedom, While the second operates directly on the degrees of freedom. Both approaches aim to introduce minimal changes to the original ParFlow code. In an AMR framework, the FD method taken by ParFlow will require modifications to correctly deal with different size elements. Mixed finite elements (MFE) are on the other hand better suited for the usage of AMR. It is known that the cell centered FD method used in ParFlow might be reinterpreted as a MFE discretization using Raviart-Thomas elements of lower order. We conclude this thesis presenting a block preconditioner for saddle point problems arising from a MFE on locally refined meshes. We evaluate its robustness with respect to various classes of coefficients for uniform and locally refined meshes

Discontinuous Galerkin Methods in Nanophotonics

In this thesis I present discontinuous Galerkin methods for Maxwell\u27s equations in both time- and frequency-domain. The method\u27s computational capabilities are extended by perfectly matched layers, dispersive and anisotropic materials, and sources. These techniques are applied to a study of coupling effects in split-ring resonator arrays

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Passive Aeroelastic Tailoring

The Passive Aeroelastic Tailoring (PAT) project was tasked with investigating novel methods to achieve passive aeroelastic tailoring on high aspect ratio wings. The goal of the project was to identify structural designs or topologies that can improve performance and/or reduce structural weight for high-aspect ratio wings. This project considered two unique approaches, which were pursued in parallel: through-thickness topology optimization and composite tow-steering

Signal Processing on Textured Meshes

In this thesis we extend signal processing techniques originally formulated in the context of image processing to techniques that can be applied to signals on arbitrary triangles meshes. We develop methods for the two most common representations of signals on triangle meshes: signals sampled at the vertices of a finely tessellated mesh, and signals mapped to a coarsely tessellated mesh through texture maps. Our first contribution is the combination of Lagrangian Integration and the Finite Elements Method in the formulation of two signal processing tasks: Shock Filters for texture and geometry sharpening, and Optical Flow for texture registration. Our second contribution is the formulation of Gradient-Domain processing within the texture atlas. We define a function space that handles chart discontinuities, and linear operators that capture the metric distortion introduced by the parameterization. Our third contribution is the construction of a spatiotemporal atlas parameterization for evolving meshes. Our method introduces localized remeshing operations and a compact parameterization that improves geometry and texture video compression. We show temporally coherent signal processing using partial correspondences