A Flexible Galerkin Finite Element Method (FGM) is a hybrid class of finite element methods that combine the usual continuous Galerkin method with the now popular discontinuous Galerkin method (DGM). A detailed description of the formulation of the FGM on a hyperbolic partial differential equation, as well as the data structures used in the FGM algorithm is presented. Some hp-convergence results and computational cost are included. Additionally, an a posteriori error estimate for the DGM applied to a two-dimensional hyperbolic partial differential equation is constructed. Several examples, both linear and nonlinear, indicating the effectiveness of the error estimate are included. / Ph. D.

In this dissertation we discuss bilinear immersed finite elements (IFE) for solving interface problems. The related research works can be categorized into three aspects: (1) the construction of the bilinear immersed finite element spaces; (2) numerical methods based on these IFE spaces for solving interface problems; and (3) the corresponding error analysis. All of these together form a solid foundation for the bilinear IFEs. The research on immersed finite elements is motivated by many real world applications, in which a simulation domain is often formed by several materials separated from each other by curves or surfaces while a mesh independent of interface instead of a body-fitting mesh is preferred. The bilinear IFE spaces are nonconforming finite element spaces and the mesh can be independent of interface. The error estimates for the interpolation of a Sobolev function in a bilinear IFE space indicate that this space has the usual approximation capability expected from bilinear polynomials, which is <i>O</i>(<i>h</i>²) in <i>L</i>² norm and <i>O</i>(<i>h</i>) in <i>H</i>¹ norm. Then the immersed spaces are applied in Galerkin, finite volume element (FVE) and discontinuous Galerkin (DG) methods for solving interface problems. Numerical examples show that these methods based on the bilinear IFE spaces have the same optimal convergence rates as those based on the standard bilinear finite element for solutions with certain smoothness. For the symmetric selective immersed discontinuous Galerkin method based on bilinear IFE, we have established its optimal convergence rate. For the Galerkin method based on bilinear IFE, we have also established its convergence. One of the important advantages of the discontinuous Galerkin method is its flexibility for both <i>p</i> and <i>h</i> mesh refinement. Because IFEs can use a mesh independent of interface, such as a structured mesh, the combination of a DG method and IFEs allows a flexible adaptive mesh independent of interface to be used for solving interface problems. That is, a mesh independent of interface can be refined wherever needed, such as around the interface and the singular source. We also develop an efficient selective immersed discontinuous Galerkin method. It uses the sophisticated discontinuous Galerkin formulation only around the locations needed, but uses the simpler Galerkin formulation everywhere else. This selective formulation leads to an algebraic system with far less unknowns than the immersed DG method without scarifying the accuracy; hence it is far more efficient than the conventional discontinuous Galerkin formulations. / Ph. D.

In this thesis, we present a simple and efficient \emph{a posteriori} error estimation procedure for a discontinuous finite element method applied to scalar first-order hyperbolic problems on structured and unstructured tetrahedral meshes. We present a local error analysis to derive a discontinuous Galerkin orthogonality condition for the leading term of the discretization error and find basis functions spanning the error for several finite element spaces. We describe an implicit error estimation procedure for the leading term of the discretization error by solving a local problem on each tetrahedron. Numerical computations show that the implicit \emph{a posteriori} error estimation procedure yields accurate estimates for linear and nonlinear problems with smooth solutions. Furthermore, we show the performance of our error estimates on problems with discontinuous solutions. We investigate pointwise superconvergence properties of the discontinuous Galerkin (DG) method using enriched polynomial spaces. We study the effect of finite element spaces on the superconvergence properties of DG solutions on each class and type of tetrahedral elements. We show that, using enriched polynomial spaces, the discretization error on tetrahedral elements having one inflow face, is O(h^{p+2}) superconvergent on the three edges of the inflow face, while on elements with one inflow and one outflow faces the DG solution is O(h^{p+2}) superconvergent on the outflow face in addition to the three edges of the inflow face. Furthermore, we show that, on tetrahedral elements with two inflow faces, the DG solution is O(h^{p+2}) superconvergent on the edge shared by two of the inflow faces. On elements with two inflow and one outflow faces and on elements with three inflow faces, the DG solution is O(h^{p+2}) superconvergent on two edges of the inflow faces. We also show that using enriched polynomial spaces lead to a simpler{a posterior error estimation procedure. Finally, we extend our error analysis for the discontinuous Galerkin method applied to linear three-dimensional hyperbolic systems of conservation laws with smooth solutions. We perform a local error analysis by expanding the local error as a series and showing that its leading term is O( h^{p+1}). We further simplify the leading term and express it in terms of an optimal set of polynomials which can be used to estimate the error. / Ph. D.

We present immersed discontinuous Galerkin finite element methods for one and two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to be cut by the material interface. The proposed methods use the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one fluid while on interface elements containing more than one fluid they use specially-built piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. Additional curl-free and consistency conditions are added to generate bilinear and biquadratic piecewise shape functions for two dimensional problems. We established the existence and uniqueness of one dimensional immersed finite element shape functions and existence of two dimensional bilinear immersed finite element shape functions for the velocity. The proposed methods are tested on one dimensional problems and are extended to two dimensional problems where the problem is defined on a domain split by an interface into two different media. Our methods exhibit optimal $O(h^{p+1})$ convergence rates for one and two dimensional problems. However it is observed that one of the proposed methods is not stable for two dimensional interface problems with high contrast media such as water/air. We performed an analysis to prove that our immersed Petrov-Galerkin method is stable for interface problems with high jumps across the interface. Local time-stepping and parallel algorithms are used to speed up computation. Several realistic interface problems such as ether/glycerol, water/methyl-alcohol and water/air with a circular interface are solved to show the stability and robustness of our methods. / Ph. D.

In recent years, the discontinuous Galerkin (DG) method has been successfully applied to solving hyperbolic conservation laws. Due to its compactness, high order accuracy, and versatility, the DG method has been extensively applied to convection-diffusion problems. In this dissertation, a numerical package, texttt{PHORCE}, is introduced to solve a number of convection-diffusion problems in neutral fluids and plasmas. Unstructured grids are used in order to randomize grid errors, which is especially important for complex geometries. texttt{PHORCE} is written in texttt{C++} and fully parallelized using the texttt{MPI} library. Memory optimization has been considered in this work to achieve improved efficiency. DG algorithms for hyperbolic terms are well studied. However, an accurate and efficient diffusion solver still constitutes ongoing research, especially for a nodal representation of the discontinuous Galerkin (NDG) method. An affine reconstructed discontinuous Galerkin (aRDG) algorithm is developed in this work to solve the diffusive operator using an unstructured NDG method. Unlike other reconstructed/recovery algorithms, all computations can be performed on a reference domain, which promotes efficiency in computation and storage. In addition, to the best of the authors' knowledge, this is the first practical guideline that has been proposed for applying the reconstruction algorithm on a nodal discontinuous Galerkin method. TVB type and WENO type limiters are also studied to deal with numerical oscillations in regions with strong physical gradients in state variables. A high-order positivity-preserving limiter is also extended in this work to prevent negative densities and pressure. A new interface tracking method, mass of fluid (MOF), along with its bound limiter has been proposed in this work to compute the mass fractions of different fluids over time. Hydrodynamic models, such as Euler and Navier-Stokes equations, and plasma models, such as ideal-magnetohydrodynamics (MHD) and two-fluid plasma equations, are studied and benchmarked with various applications using this DG framework. Numerical computations of Rayleigh-Taylor instability growth with experimentally relevant parameters are performed using hydrodynamic and MHD models on planar and radially converging domains. Discussions of the suppression mechanisms of Rayleigh-Taylor instabilities due to magnetic fields, viscosity, resistivity, and thermal conductivity are also included. This work was partially supported by the US Department of Energy under grant number DE-SC0016515. The author acknowledges Advanced Research Computing at Virginia Tech for providing computational resources and technical support that have contributed to the results reported within this work. URL: http://www.arc.vt.edu / Doctor of Philosophy / High-energy density (HED) plasma science is an important area in studying astrophysical phenomena as well as laboratory phenomena such as those applicable to inertial confinement fusion (ICF). ICF plasmas undergo radial compression, with an aim of achieving fusion ignition, and are subject to a number of hydrodynamic instabilities that can significantly alter the implosion and prevent sufficient fusion reactions. An understanding of these instabilities and their mitigation mechanisms is important allow for a stable implosion in ICF experiments. This work aims to provide a high order accurate and robust numerical framework that can be used to study these instabilities through simulations. The first half of this work aims to provide a detailed description of the numerical framework, texttt{PHORCE}. texttt{PHORCE} is a high order numerical package that can be used in solving convection-diffusion problems in neutral fluids and plasmas. Outstanding challenges exist in simulating high energy density (HED) hydrodynamics, where very large gradients exist in density, temperature, and transport coefficients (such as viscosity), and numerical instabilities arise from these region if there is no intervention. These instabilities may lead to inaccurate results or cause simulations to fail, especially for high-order numerical methods. Substantial work has been done in texttt{PHORCE} to improve its robustness in dealing with numerical instabilities. This includes the implementation and design of several high-order limiters. An novel algorithm is also proposed in this work to solve the diffusion term accurately and efficiently, which further enriches the physics that texttt{PHORCE} can investigate. The second half of this work involves rigorous benchmarks and experimentally relevant simulations of hydrodynamic instabilities. Both advection and diffusion solvers are well verified through convergence studies. Hydrodynamic and plasma models implemented are also validated against results in existing literature. Rayleigh-Taylor instability growth with experimentally relevant parameters are performed on both planar and radially converging domains. Although this work is motivated by physics in HED hydrodynamics, the emphasis is placed on numerical models that are generally applicable across a wide variety of fields and disciplines.

A careful study of plasma-material interactions is essential to understand and improve the operation of devices where plasma contacts a wall such as plasma thrusters, fusion devices, spacecraft-environment interactions, to name a few. This work aims to advance our understanding of fundamental plasma processes pertaining to plasma-material interactions, sheath physics, and kinetic instabilities through theory and novel numerical simulations. Key contributions of this work include (i) novel continuum kinetic algorithms with novel boundary conditions that directly discretize the Vlasov/Boltzmann equation using the discontinuous Galerkin method, (ii) fundamental studies of plasma sheath physics with collisions, ionization, and physics-based wall emission, and (iii) theoretical and numerical studies of the linear growth and nonlinear saturation of the kinetic Weibel instability, including its role in plasma sheaths. The continuum kinetic algorithm has been shown to compare well with theoretical predictions of Landau damping of Langmuir waves and the two-stream instability. Benchmarks are also performed using the electromagnetic Weibel instability and excellent agreement is found between theory and simulation. The role of the electric field is significant during nonlinear saturation of the Weibel instability, something that was not noted in previous studies of the Weibel instability. For some plasma parameters, the electric field energy can approach magnitudes of the magnetic field energy during the nonlinear phase of the Weibel instability. A significant focus is put on understanding plasma sheath physics which is essential for studying plasma-material interactions. Initial simulations are performed using a baseline collisionless kinetic model to match classical sheath theory and the Bohm criterion. Following this, a collision operator and volumetric physics-based source terms are introduced and effects of heat flux are briefly discussed. Novel boundary conditions are developed and included in a general manner with the continuum kinetic algorithm for bounded plasma simulations. A physics-based wall emission model based on first principles from quantum mechanics is self-consistently implemented and demonstrated to significantly impact sheath physics. These are the first continuum kinetic simulations using self-consistent, wall emission boundary conditions with broad applicability across a variety of regimes. / Ph. D. / An understanding of plasma physics is vital for problems on a wide range of scales: from large astrophysical scales relevant to the formation of intergalactic magnetic fields, to scales relevant to solar wind and space weather, which poses a significant risk to Earth’s power grid, to design of fusion devices, which have the potential to meet terrestrial energy needs perpetually, and electric space propulsion for human deep space exploration. This work aims to further our fundamental understanding of plasma dynamics for applications with bounded plasmas. A comprehensive understanding of theory coupled with high-fidelity numerical simulations of fundamental plasma processes is necessary, this then can be used to improve improve the operation of plasma devices. There are two main thrusts of this work. The first thrust involves advancing the state-of-the-art in numerical modeling. Presently, numerical simulations in plasma physics are typically performed either using kinetic models such as particle-in-cell, where individual particles are tracked through a phase-space grid, or using fluid models, where reductions are performed from kinetic physics to arrive at continuum models that can be solved using well-developed numerical methods. The novelty of the numerical modeling is the ability to perform a complete kinetic calculation using a continuum description and evolving a complete distribution function in phase-space, thus resolving kinetic physics with continuum numerics. The second thrust, which is the main focus of this work, aims to advance our fundamental understanding of plasma-wall interactions as applicable to real engineering problems. The continuum kinetic numerical simulations are used to study plasma-material interactions and their effects on plasma sheaths. Plasma sheaths are regions of positive space charge formed everywhere that a plasma comes into contact with a solid surface; the charge inequality is created because mobile electrons can quickly exit the domain. A local electric field is self-consistently created which accelerates ions and retards electrons so the ion and electron fluxes are equalized. Even though sheath physics occurs on micro-scales, sheaths can have global consequences. The electric field accelerates ions towards the wall which can cause erosion of the material. Another consequence of plasma-wall interaction is the emission of electrons. Emitted electrons are accelerated back into the domain and can contribute to anomalous transport. The novel numerical method coupled with a unique implementation of electron emission from the wall is used to study plasma-wall interactions. While motivated by Hall thrusters, the applicability of the algorithms developed here extends to a number of other disciplines such as semiconductors, fusion research, and spacecraft-environment interactions.

This thesis develops an immersed finite element (IFE) space for numerical simulations arising from beam design with multiple materials. This IFE space is based upon meshes that can be independent of interface of the materials used to form a beam. Both the forward and inverse problems associated with the beam equation are considered. The order of accuracy of this IFE space is numerically investigated from the point of view of both the interpolation and finite element solution of the interface boundary value problems. Both single and multiple interfaces are considered in our numerical simulation. The results demonstrate that this IFE space has the optimal order of approximation capability. / Master of Science

In this work, we develop a method for solving viscoelastic fluid flows using the Navier-Stokes equations coupled with the Oldroyd-B model. We solve the Navier-Stokes equations in skew-symmetric form using the mixed finite element method, and we solve the Oldroyd-B model using the discontinuous Galerkin method. The Crank-Nicolson scheme is used for the temporal discretization of the Navier-Stokes equations in order to achieve a second-order accuracy in time, while the optimal third-order total-variation diminishing Runge-Kutta scheme is used for the temporal discretization of the Oldroyd-B equation. The overall accuracy in time is therefore limited to second-order due to the Crank-Nicolson scheme; however, a third-order Runge-Kutta scheme is implemented for greater stability over lower order Runge-Kutta schemes. We test our numerical method using the 2D cavity flow benchmark problem and compare results generated with those found in literature while discussing the influence of mesh refinement on suppressing oscillations in the polymer stress. / Master of Science / Viscoelastic fluids are a type of non-Newtonian fluid of great importance to the study of fluid flows. Such fluids exhibit both viscous and elastic behaviors. We develop a numerical method to solve the partial differential equations governing viscoelastic fluid flows using various finite element methods. Our method is then validated using previous numerical results in literature.

Les changements rapides et profonds sont des plus en plus fréquents, tant dans les milieux de travail que dans la société. Ces changements rapides et profonds, nommés changements discontinus, sont connus pour être éprouvants pour le bien-être psychologique des individus. La littérature a proposé que les changements discontinus organisationnels et sociaux soient éprouvants parce qu’ils perturbent l’identité collective, qui fournit aux individus un cadre de référence dans lequel ils comprennent tant leur monde social qu’eux-mêmes. En réponse à un changement discontinu, l’identité collective souffrirait d’une baisse de clarté, où les individus se questionnent à savoir « qui ils sont » dans le contexte de leur groupe social. De récentes études confirment qu’un changement social discontinu provoque une baisse de clarté de l’identité collective. Toutefois, le lien entre les changements discontinus et la clarté d’une identité collective nécessite un soutien empirique robuste puisqu’aucune étude n’a manipulé expérimentalement un réel changement vécu par un groupe social. De plus, il reste à vérifier si la baisse du niveau de clarté de l’identité collective en réponse à un changement social discontinu est répliquée empiriquement sur le terrain. Le premier objectif de cette thèse est donc de vérifier l’effet d’un réel changement discontinu sur la clarté de l’identité collective d’un groupe social. Par ailleurs, les processus psychologiques qui déterminent comment les individus rétablissent la clarté de leur identité collective à la suite d’un changement discontinu demeurent à ce jour inconnus. La littérature en psychologie sociale soutient que deux processus de comparaisons contribuent à la construction de l’identité collective, soit les comparaisons sociales et les comparaisons temporelles. Il semblerait que les comparaisons temporelles soient plus prévalentes dans un contexte de changement discontinu, mais leur rôle pour rétablir la clarté de l’identité collective n’a pas été vérifié. Le second objectif consiste à vérifier si le fait d’effectuer des comparaisons temporelles est un processus psychologique qui rétablit la clarté de l’identité collective à la suite d’un changement discontinu. Cinq études réparties en deux articles ont été exécutées pour répondre à ces objectifs. Le premier article comble les lacunes soulevées dans la littérature en présentant un nouveau paradigme expérimental : le paradigme de groupes de travail Lego (PGTL). Ce paradigme expérimental simule un groupe de travail et expose ses membres à un réel changement discontinu, opérationnalisé en tant qu’un changement inattendu des valeurs qui orientent les objectifs de travail. Trois études testent l’hypothèse que l’introduction d’un changement discontinu causera une diminution du niveau de clarté de l’identité collective du groupe de travail. L’étude 1 simule un changement discontinu en transformant subitement les valeurs du groupe de collaboration à des valeurs de compétition. L’étude 2 réplique les résultats de l’étude 1 avec différente paire de valeurs opposées, soit l’efficience et l’innovation. La troisième étude réplique les résultats et la méthodologie de l’étude 1 avec un grand échantillon, ce qui permet de contrôler statistiquement pour la non-indépendance des observations. Les trois études confirment l’hypothèse que l’introduction d’un changement discontinu cause une diminution du niveau de clarté de l’identité collective du groupe de travail. Le deuxième article contient deux études qui se déroulent auprès d’Américains, dans le contexte du changement d’administration présidentielle en 2016. La première étude vérifie dans un premier temps si la clarté de l’identité collective est diminuée à la suite d’un changement discontinu sur le terrain (hypothèse 1). Dans un deuxième temps, une intervention utilisant des comparaisons temporelles est testée pour vérifier si elle permet de rétablir la clarté de l’identité collective (hypothèse 2a). Le degré d’efficacité de l’intervention utilisant des comparaisons temporelles est comparé à une intervention utilisant des comparaisons sociales, une intervention utilisant des comparaisons sociales et temporelles et une condition contrôle. La deuxième étude vérifie si l’intervention utilisant des comparaisons temporelles rétablit la clarté de l’identité collective au-delà de l’effet du passage du temps (hypothèse 2b) et au-delà des autres interventions. Les résultats confirment la diminution de clarté de l’identité collective à la suite de l’élection, et le rôle de comparaisons temporelles pour rétablir la clarté de l’identité collective. / Rapid and profound changes are increasingly common, both in the workplace and in society. These rapid and profound changes to social groups, called discontinuous changes, are known to have deleterious effects on the psychological well-being of individuals. Literature has proposed that discontinuous organizational and social changes are challenging because they disrupt individuals’ collective identity, which provides them with a meaningful frame of reference in which they understand their social environment and themselves. It has been proposed that during discontinuous changes, collective identity suffers from a decrease in clarity, where individuals question who they are in the context of their social group. Recent studies confirm that discontinuous social change causes a decrease in collective identity clarity. However, the link between discontinuous changes and collective identity clarity remains tentative since no study has experimentally manipulated a real experienced change in a social group. Finally, it remains to be seen whether the decrease in collective identity clarity following discontinuous social change is empirically replicated in the field. The first goal is therefore to provide robust empirical support to the proposition that discontinuous change causes a decrease of collective identity clarity, in a context of real experienced change. In addition, the psychological processes that determine how individuals restore collective identity clarity following discontinuous change remain unknown to this day. Literature in social psychology argues that there are two processes of comparison fundamental to the construction of collective identity, namely social comparisons and temporal comparisons. Temporal comparisons appear to be more prevalent in a context of discontinuous change, but their role in restoring collective identity clarity has not been verified. The second goal of this thesis is to verify whether temporal comparisons are a psychological process that restores collective identity clarity following a discontinuous change. Five studies divided into two articles were carried out to meet both goals. The first article fills the gaps raised in previous literature by presenting a new experimental paradigm: the Lego Workgroup Paradigm (LWP). This experimental paradigm simulates a work group and exposes its members to a real experienced discontinuous change, operationalized as an unexpected change in the values that guide the work objectives. Three studies test the hypothesis that the introduction of a discontinuous change will reduce levels of collective identity clarity related to the workgroup. Study 1 simulates a discontinuous change transforming the values in a sudden way from collaborative values to competition values. Study 2 replicates the results of Study 1 with different opposing values of efficiency and innovation. The third study replicates the results and methodology of Study 1 with a large sample that allows statistical control for the non-independence of observations. The three studies support the hypothesis that the introduction of a discontinuous change causes a decrease in collective identity clarity related to the work group. The second article contains two studies conducted with Americans, in the context of the change of presidential administration in 2016. The first study aims to validate whether the clarity of the collective identity is diminished as a result of a discontinuous change in a field setting (hypothesis 1). Then, an intervention using temporal comparisons is tested to verify if it restores collective identity clarity (hypothesis 2a). The degree of effectiveness of the intervention using temporal comparisons is contrasted with an intervention using social comparisons, an intervention using social and temporal comparisons and a control condition. The second study verifies whether the intervention using temporal comparisons restores collective identity clarity beyond the effect of the passage of time (hypothesis 2b), and beyond other interventions. The results confirm the decrease in collective identity clarity as a result of the election’s outcome and the role of temporal comparisons to restore the collective identity clarity.

We consider a family of variational time discretizations that generalizes discontinuous Galerkin (dG) and continuous Galerkin-Petrov (cGP) methods. In addition to variational conditions the methods also contain collocation conditions in the time mesh points. The single family members are characterized by two parameters that represent the local polynomial ansatz order and the number of non-variational conditions, which is also related to the global temporal regularity of the numerical solution. Moreover, with respect to Dahlquist’s stability problem the variational time discretization (VTD) methods either share their stability properties with the dG or the cGP method and, hence, are at least A-stable. With this thesis, we present the first comprehensive theoretical study of the family of VTD methods in the context of non-stiff and stiff initial value problems as well as, in combination with a finite element method for spatial approximation, in the context of parabolic problems. Here, we mainly focus on the error analysis for the discretizations. More concrete, for initial value problems the pointwise error is bounded, while for parabolic problems we rather derive error estimates in various typical integral-based (semi-)norms. Furthermore, we show superconvergence results in the time mesh points. In addition, some important concepts and key properties of the VTD methods are discussed and often exploited in the error analysis. These include, in particular, the associated quadrature formulas, a beneficial postprocessing, the idea of cascadic interpolation, connections between the different VTD schemes, and connections to other classes of methods (collocation methods, Runge-Kutta-like methods). Numerical experiments for simple academic test examples are used to highlight various properties of the methods and to verify the optimality of the proven convergence orders.:List of Symbols and Abbreviations Introduction I Variational Time Discretization Methods for Initial Value Problems 1 Formulation, Analysis for Non-Stiff Systems, and Further Properties 1.1 Formulation of the methods 1.1.1 Global formulation 1.1.2 Another formulation 1.2 Existence, uniqueness, and error estimates 1.2.1 Unique solvability 1.2.2 Pointwise error estimates 1.2.3 Superconvergence in time mesh points 1.2.4 Numerical results 1.3 Associated quadrature formulas and their advantages 1.3.1 Special quadrature formulas 1.3.2 Postprocessing 1.3.3 Connections to collocation methods 1.3.4 Shortcut to error estimates 1.3.5 Numerical results 1.4 Results for affine linear problems 1.4.1 A slight modification of the method 1.4.2 Postprocessing for the modified method 1.4.3 Interpolation cascade 1.4.4 Derivatives of solutions 1.4.5 Numerical results 2 Error Analysis for Stiff Systems 2.1 Runge-Kutta-like discretization framework 2.1.1 Connection between collocation and Runge-Kutta methods and its extension 2.1.2 A Runge-Kutta-like scheme 2.1.3 Existence and uniqueness 2.1.4 Stability properties 2.2 VTD methods as Runge-Kutta-like discretizations 2.2.1 Block structure of A VTD 2.2.2 Eigenvalue structure of A VTD 2.2.3 Solvability and stability 2.3 (Stiff) Error analysis 2.3.1 Recursion scheme for the global error 2.3.2 Error estimates 2.3.3 Numerical results II Variational Time Discretization Methods for Parabolic Problems 3 Introduction to Parabolic Problems 3.1 Regularity of solutions 3.2 Semi-discretization in space 3.2.1 Reformulation as ode system 3.2.2 Differentiability with respect to time 3.2.3 Error estimates for the semi-discrete approximation 3.3 Full discretization in space and time 3.3.1 Formulation of the methods 3.3.2 Reformulation and solvability 4 Error Analysis for VTD Methods 4.1 Error estimates for the l th derivative 4.1.1 Projection operators 4.1.2 Global L2-error in the H-norm 4.1.3 Global L2-error in the V-norm 4.1.4 Global (locally weighted) L2-error of the time derivative in the H-norm 4.1.5 Pointwise error in the H-norm 4.1.6 Supercloseness and its consequences 4.2 Error estimates in the time (mesh) points 4.2.1 Exploiting the collocation conditions 4.2.2 What about superconvergence!? 4.2.3 Satisfactory order convergence avoiding superconvergence 4.3 Final error estimate 4.4 Numerical results Summary and Outlook Appendix A Miscellaneous Results A.1 Discrete Gronwall inequality A.2 Something about Jacobi-polynomials B Abstract Projection Operators for Banach Space-Valued Functions B.1 Abstract definition and commutation properties B.2 Projection error estimates B.3 Literature references on basics of Banach space-valued functions C Operators for Interpolation and Projection in Time C.1 Interpolation operators C.2 Projection operators C.3 Some commutation properties C.4 Some stability results D Norm Equivalences for Hilbert Space-Valued Polynomials D.1 Norm equivalence used for the cGP-like case D.2 Norm equivalence used for final error estimate Bibliography