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This work is concerned with the derivation of adaptive methods for discontinuous Galerkin approximations of linear fourth order elliptic and parabolic partial differential equations. Adaptive methods are usually based on a posteriori error estimates. To this end, a new residual-based a posteriori error estimator for discontinuous Galerkin approximations to the biharmonic equation with essential boundary conditions is presented. The estimator is shown to be both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. The reliability bound is based on a new recovery operator, which maps discontinuous finite element spaces to conforming finite element spaces (of two polynomial degrees higher), consisting of triangular or quadrilateral Hsieh-Clough-Tocher macroelements. The efficiency bound is based on bubble function techniques. The performance of the estimator within an h-adaptive mesh refinement procedure is validated through a series of numerical examples, verifying also its asymptotic exactness. Some remarks on the question of proof of convergence of adaptive algorithms for discontinuous Galerkin for fourth order elliptic problems are also presented. Furthermore, we derive a new energy-norm a posteriori error bound for an implicit Euler time-stepping method combined with spatial discontinuous Galerkin scheme for linear fourth order parabolic problems. A key tool in the analysis is the elliptic reconstruction technique. A new challenge, compared to the case of conforming finite element methods for parabolic problems, is the control of the evolution of the error due to non-conformity. Based on the error estimators, we derive an adaptive numerical method and discuss its practical implementation and illustrate its performance in a series of numerical experiments.

We consider both the a priori and a posteriori error analysis and hp-adaptation strategies for discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched polynomial degrees. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution, exploiting duality based arguments.;The a priori error analysis is carried out in two settings. In the first, full orientation of elements is allowed but only (possibly high-order) isotropic polynomial degrees considered; our analysis, therefore, extends previous results, where only finite element spaces comprising piecewise linear polynomials were considered, by utilizing techniques from tensor analysis. In the second case, anisotropic polynomial degrees are allowed, but the elements are assumed to be axiparallel; we thus apply previously known interpolation error results to the goal-oriented setting.;Based on our a posteriori error bound we first design and implement an adaptive anisotropic h-refinement algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement, chosen on a competitive basis requiring the solution of local problems. The superiority of the proposed algorithm in comparison with a standard h-isotropic mesh refinement algorithm and a Hessian based h-anisotropic adaptive procedure is illustrated by a series of numerical experiments. We then describe a fully hp -adaptive algorithm, once again using a competitive refinement approach, which, numerical experiments reveal, offers considerable improvements over both a standard hp-isotropic refinement algorithm and an h-anisotropic/p-isotropic adaptive procedure.

The primary aim of this thesis has been to study examples of the application of delay differential equations to both physiology and epidemiology. As such, the thesis has two main strands. The physiological application is represented by mathematical models of the glucose-insulin interaction in humans. We provide a detailed introduction to recent and current literature associated with this area, together with an overview of the physiological processes involved. Two systems explicitly incorporating a discrete delay are proposed and positivity and boundedness of solutions to these models are established. Sufficient conditions for global stability of the steady state of both systems are derived using both Lyapunov methods and comparison principles. Physiological interpretations of the analysis are provided. The simpler of the two models is then extended to represent a person being given periodic infusions of both insulin and glucose. Positivity of solutions of this system is established and the existence of a positive periodic solution is proved using the coincidence degree theory method. The epidemiological application of delay differential equations is represented by mathematically modelling the transmission dynamics of tuberculosis. A brief overview of the current impact of the disease is given and some of the problems public health officials face in combatting it are discussed. A summary of work in the literature on this subject is provided. The effect of migration on the spread of tuberculosis is considered. Patch type models consisting of just two patches are proposed, each patch considered to be a country. We allow for the possibility that migration between two countries is often more one way than the other so diffusion is not the discrete analogue of Fickian diffusion. Conditions for both local and global stability of the disease-free steady state are determined using a variety of methods. A model with a continuous representation of space incorporating Fickian diffusion is then proposed. This is assumed to be more appropriate for various animal species than for humans. The possibility of a travelling wave-front solution is investigated and the minimum speed of such a solution is determined. Numerical simulations support these results. Finally the model is adapted to incorporate the tendency to move towards a focal point. Using numerical simulations, the effect of random dispersal and purposeful movement towards a focal point are investigated.

This Thesis demonstrates advanced new discretisation technologies that improve the accuracy and stability of the discontinuous Galerkin finite element method applied to the Boltzmann Transport Equation, describing the advective transport of neutral particles such as photons and neutrons within a domain. The discontinuous Galerkin method in its standard form is susceptible to oscillation detrimental to the solution. The discretisation schemes presented in this Thesis enhance the basic form with linear and non-linear Petrov Galerkin methods that remove these oscillations. The new schemes are complemented by an adjoint-based error recovery technique that improves the standard solution when applied to goal-based functional and eigenvalue problems. The chapters in this Thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 outlines the Thesis and contains a brief literature review. Chapter 2 introduces the underlying space-angle discretisation method used in the work, and discusses a series of potential modifications to the standard discontinuous Galerkin method. These differ in how the upwinding is performed on the element boundary, and comprise an upwind-average method, a Petrov-Galerkin method that removes oscillation by adding artificial diffusion internal to an element and a more sophisticated Petrov-Galerkin scheme that adds dissipation in the coupling between each element. These schemes are tested in one-dimension and Taylor analysis of their convergence rate is included. The chapter concludes with selection of one of the schemes to be developed in the next part of the Thesis. Chapter 3 develops the selected method extending it to multi-dimensions. The result is a new discontinuous Petrov-Galerkin method that is residual based and removes unwanted oscillation from the transport solution by adding numerical dissipation internal to an element. The method uses a common length scale in the upwind term for all elements. This is not always satisfactory, however, as it gives the same magnitude and type of dissipation everywhere in the domain. The chapter concludes by recommending some form of non-linearity be included to address this issue. Chapter 4 adds non-linearity to the scheme. This projects the streamline direction, in which the dissipation acts, onto the solution gradient direction. It defines locally the optimal amount of dissipation needed in the discretisation. The non-linear scheme is tested on a variety of steady-state and time-dependent transport problems. Chapters 5, 6 and 7 develop an adjoint-based error measure to complement the scheme in functional and eigenvalue problems. This is done by deriving an approximation to the error in the the bulk functional or eigenvalue, and then removing it from the calculated value in a post-process defect iteration. This is shown to dramatically accelerate mesh convergence of the goal-based functional or eigenvalue. Chapter 8 concludes the Thesis with recommendations for a further plan of work.

Στο πρώτο κεφάλαιο περιγράφονται οι άμεσες μέθοδοι Runge – Kutta και ο προτεινόμενος ελεγκτής που είναι τύπου PI. Όταν το μέγεθος βήματος περιορίζεται από την αριθμητική ευστάθεια, ένα δυναμικό μοντέλο πρέπει να χρησιμοποιηθεί. Ένα τέτοιο μοντέλο παρήχθη και επαληθεύτηκε αριθμητικά για άμεσες μεθόδους Runge – Kutta. Εδώ περιγράφεται αυτό το δυναμικό μοντέλο. Στο δεύτερο κεφάλαιο περιγράφονται το πρόβλημα της επιλογής μεγέθους βήματος στα έμμεσα σχήματα Runge – Kutta και αναλύεται από μια άποψη ελέγχου ανατροφοδότησης. Οι ιδιότητες του νέου μοντέλου και της βελτιωμένης απόδοσης του νέου ελέγχου σφάλματος περιγράφονται χρησιμοποιώντας και ανάλυση και αριθμητικά παραδείγματα. Στο τρίτο κεφάλαιο αναλύεται και υλοποιείται σε περιβάλλον Μathematica η μη γραμμική διαφορική εξίσωση van der Ρol για μια σειρά από διαφορετικές τιμές της παραμέτρου ε. Επιπλέον σ’ αυτό το κεφάλαιο μελετάται η συμπεριφορά του συστήματος με την μέθοδο Runge-Kutta και με βάση τον ολοκληρωμένο αλγόριθμο του P ελέγχου βήματος. Στο τέταρτο κεφάλαιο περιγράφονται οι βασικές μέθοδοι για την επίλυση μη δύσκαμπτων συστημάτων συνήθων διαφορικών εξισώσεων από χαμηλές σε μεσαίες ανοχές. Εδώ δείχνεται πώς κατασκευάζονται μερικά ζεύγη χαμηλής τάξης χρησιμοποιώντας εργαλεία από την υπολογιστική άλγεβρα. Εστιάζεται η προσοχή μας πάνω σε μεθόδους που εξοπλίζονται με ανίχνευση τοπικού σφάλματος (για προσαρμοστικότητα στο μέγεθος βήματος) και με τη δυνατότητα να ανιχνευθεί η δυσκαμψία. Στο πέμπτο κεφάλαιο υλοποιείται σε περιβάλλον Mathematica η σύγκριση δυο αλγορίθμων ελέγχου (P και PI) του βήματος στην μη γραμμική διαφορική εξίσωση van der Pol υλοποιημένη σε RKclassic και RKdopri μέθοδο Στο έκτο κεφάλαιο μελετάται ένα πραγματικό σύστημα της μορφής y'=Α*y με βάση την μεθοδολογία που το προσομοιώνει η μέθοδος Runge-Kutta σε λογισμικό περιβάλλον Mathematica. Στο τελευταίο (έβδομο) κεφάλαιο γίνεται η σύγκριση του πραγματικού συστήματος και του προσομοιωμένου PI έλεγχου βήματος για τη μέθοδο Runge-Kutta. / -