We present the analysis for the higher order continuous Galerkin−Petrov (cGP) time discretization schemes in combination with the one-level local projection stabilization in space applied to time-dependent convection-diffusion-reaction problems. Optimal a priori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous Galerkin−Petrov and discontinuous Galerkin time discretization schemes will be given.

As traditional communication technology we use in our day-to-day life reaches its limitations, the international community searches for new methods to communicate information. One such novel approach is the so-called molecular communication system. During the last few decades, molecular communication systems become more and more popular. The main difference between traditional communication and molecular communication systems is that in the latter, information transfer occurs through chemical means, most often between microorganisms. This process already happens all around us naturally, for example, in the human body. Even though the molecular communication topic is attractive to researchers, and a lot of theoretical results are available - one cannot claim the same about the practical use of molecular communication. As for experimental results, a few studies have been done on the macroscale, but investigations at the micro- and nanoscale ranges are still lacking because they are a challenging task. In this work, a self-contained introduction of the underlying theory of molecular communication is provided, which includes knowledge from different areas such as biology, chemistry, communication theory, and applied mathematics. Two numerical methods are implemented for three well-studied partial differential equations of the MC field where advection, diffusion, and the reaction are taken into account. Numerical results for test cases in one and three dimensions are presented and discussed in detail. Conclusions and essential analytical and numerical future directions are then drawn.

The objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von K\'rm\'{a}n-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented. We used two approaches to study the responses of shells to a primary resonance: a $direct$ approach and a $discretization$ approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency- and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared. For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating $2n+(2n)^2$ nonlinear first-order ordinary-differential equations simultaneously, where $n$ is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results. The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown. Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo symmetry-breaking, saddle node, and period-doubling bifurcations. / Ph. D.

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Porosity is known to be one of the primary detrimental factors controlling fatigue life and total elongation of several cast alloy components. The two main aims of this work are to examine pore nucleation and growth effects for predicting gas microporosity and to study the physics of bifilm dynamics to gain understanding in the role of bifilms in producing defects and the mechanisms of defect creation. In the second chapter of this thesis, an innovative technique, based on the combination of a set of conservation equations that solves the transport phenomena during solidification at the macro-scale and the hydrogen diffusion into the pores at the micro-scale, was used to quantify the amount of gas microporosity in A356 alloy castings. The results were compared with published experimental data. In the reminder of this work, the Immersed Element-Free Galerkin method (IEFGM) is presented and it was used to study the physics of bifilm dynamics. The IEFGM is an extension of the Immersed Finite Element method (IFEM) developed by Zhang et al. [50] and it is an attractive technique for simulating FSI problems involving highly deformable bifilm-like solids.

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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

In this thesis, we present new superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional hyperbolic problems. We investigate the superconvergence properties of the DG method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We study the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements. Superconvergence is described for structured and unstructured meshes. We show that the DG solution is O(hp+1) superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three p- degree polynomial spaces. For triangles having two outflow edges the finite element error is O(hp+1) superconvergent at the end points of the inflow edge for an augmented space of degree p. Furthermore, we discovered additional mesh-orientation dependent superconvergence points in the interior of triangles. The dependence of these points on orientation is explicitly given. We also established a global superconvergence result on meshes consisting of triangles having one inflow and one outflow edges. Applying a local error analysis, we construct simple, efficient and asymptotically correct a posteriori error estimates for discontinuous finite element solutions of hyperbolic problems on triangular meshes. A posteriori error estimates are needed to guide adaptive enrichment and to provide a measure of solution accuracy for any numerical method. We develop an inexpensive superconvergence-based a posteriori error estimation technique for the DG solutions of conservation laws. We explicitly write the basis functions for the error spaces corresponding to several finite element solution spaces. The leading term of the discretization error on each triangle is estimated by solving a local problem where no boundary conditions are needed. The computed error estimates are shown to converge to the true error under mesh refinement in smooth solution regions. We further present a numerical study of superconvergence properties for the DG method applied to time-dependent convection problems. We also construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on general unstructured meshes. The global superconvergence results are numerically confirmed. Finally, the a posteriori error estimates are tested on several linear and nonlinear problems to show their efficiency and accuracy under mesh refinement. / Ph. D.