This thesis is devoted to the numerical analysis of mesh-free methods and, in particular, to the study of the possible advantages of the EFG (Element Free Galerkin) mesh-free method against the well-known FE (Finite Element) method. More precisely, the EFG method and the FE method behavior are compared in two particular interesting problems: (1) analysis of volumetric locking in mechanical problems and (2) accurate resolution of transient convection dominated problems. In both cases the good properties and possibilities of mesh-free methods become apparent. However, in several situations the FE method is still more competitive: for instance, the computation of the FE shape functions and its integrals are less costly, and essential boundary conditions can be easily imposed. Thus, in order to take advantage of the good properties of both methods, a mixed interpolation combining FE and EFG is proposed. This formulation can be applied in two useful situations: (i) enrichment of finite elements with EFG, and (ii) coupling of FE and EFG. An a priori error estimate for the first one is presented and proved. Several examples show the applicability of the mixed interpolation in adaptive computations. / Aquesta tesi està dedicada a l'anàlisi numèrica dels mètodes sense malla i, en particular, a l'estudi dels possibles avantatges del mètode EFG (Element Free Galerkin) davant del ben conegut MEF (Mètode dels Elements Finits). Concretament, es comparen el mètode EFG i el MEF en dos problemes concrets d'interès: (1) l'anàlisi del bloqueig volumètric en problemes mecànics i (2) la resolució precisa de problemes transitoris amb convecció dominant. Les bones propietats i possibilitats dels mètodes sense malla es fan evidents en tots dos casos.Tot i així, en varis aspectes el MEF resulta més competitiu: per exemple, el càlcul de les funcions de forma i de les seves integrals es menys costós, i les condicions de contorn essencials es poden imposar fàcilment. Amb l'objectiu d'aprofitar les bones qualitats dels dos mètodes, es proposa una interpolació mixta combinant elements finits y EFG, aplicable en dues situacions: (i) enriquiment d'elements finits amb EFG i (ii) acoblament d'elements finits i EFG. Per al primer cas, es presenta i demostra una cota a priori de l'error. L'aplicabilitat d'aquesta interpolació mixta en processos adaptatius es mostra amb varis exemples. / Esta tesis está dedicada al análisis numérico de los métodos sin malla y, en particular, al estudio de las posibles ventajas del método EFG (Element Free Galerkin) frente al bien conocido MEF (Método de los Elementos Finitos). Concretamente, se comparan el método EFG y el MEF en dos problemas concretos de interés: (1) el análisis del bloqueo volumétrico en problemas mecánicos y (2) la resolución precisa de problemas transitorios con convección dominante. Las buenas propiedades y posibilidades de los métodos sin malla se hacen evidentes en ambos casos.Sin embargo, en varios aspectos el MEF resulta más competitivo: por ejemplo, el cálculo de las funciones de forma y sus integrales es menos costoso, y las condiciones de contorno esenciales se pueden imponer fácilmente. Con el objetivo de aprovechar las buenas cualidades de ambos métodos, se propone una interpolación mixta combinando elementos finitos y EFG, aplicable en dos situaciones: (i) enriquecimiento de elementos finitos con EFG, y (ii) acoplamiento de elementos finitos y EFG. Para el primer caso, se presenta y demuestra una cota a priori del error. La aplicabilidad de esta interpolación mixta en procesos adaptativos se muestra con varios ejemplos.

In this study, the Element-Free Galerkin (EFG) method has been used for the analysis of plane stress problems. A computer program has been developed by using FORTRAN language. The moving least squares (MLS) approximation has been used in generating shape functions. The results obtained by the EFG method have been compared with analytical solution and the numerical results obtained by MSC. Patran/Nastran. The comparisons show that the mesh free method gives more accurate results than the finite element approximation with less computational effort.

Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided. / South Africa

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Mathematical definitions provide a precise, unambiguous way to formulate concepts. They also provide a common language between disciplines. Thus, they are the basis for a well-founded scientific discussion. In addition, mathematical definitions allow for deeper insights into the defined subject based on mathematical theorems that are incontrovertible under the given definition. Besides their value in mathematics, mathematical definitions are indispensable in other sciences like physics, chemistry, and computer science. In computer science, they help to derive the expected behavior of a computer program and provide guidance for the design and testing of software. Therefore, mathematical definitions can be used to design and implement advanced algorithms. One class of widely used algorithms in computer science is the class of particle-based algorithms, also known as particle methods. Particle methods can solve complex problems in various fields, such as fluid dynamics, plasma physics, or granular flows, using diverse simulation methods, including Discrete Element Methods (DEM), Molecular Dynamics (MD), Reproducing Kernel Particle Methods (RKPM), Particle Strength Exchange (PSE), and Smoothed Particle Hydrodynamics (SPH). Despite the increasing use of particle methods driven by improved computing performance, the relation between these algorithms remains formally unclear. In particular, particle methods lack a unifying mathematical definition and precisely defined terminology. This prevents the determination of whether an algorithm belongs to the class and what distinguishes the class. Here we present a rigorous mathematical definition for determining particle methods and demonstrate its importance by applying it to several canonical algorithms and those not previously recognized as particle methods. Furthermore, we base proofs of theorems about parallelizability and computational power on it and use it to develop scientific computing software. Our definition unified, for the first time, the so far loosely connected notion of particle methods. Thus, it marks the necessary starting point for a broad range of joint formal investigations and applications across fields.:1 Introduction 1.1 The Role of Mathematical Definitions 1.2 Particle Methods 1.3 Scope and Contributions of this Thesis 2 Terminology and Notation 3 A Formal Definition of Particle Methods 3.1 Introduction 3.2 Definition of Particle Methods 3.2.1 Particle Method Algorithm 3.2.2 Particle Method Instance 3.2.3 Particle State Transition Function 3.3 Explanation of the Definition of Particle Methods 3.3.1 Illustrative Example 3.3.2 Explanation of the Particle Method Algorithm 3.3.3 Explanation of the Particle Method Instance 3.3.4 Explanation of the State Transition Function 3.4 Conclusion 4 Algorithms as Particle Methods 4.1 Introduction 4.2 Perfectly Elastic Collision in Arbitrary Dimensions 4.3 Particle Strength Exchange 4.4 Smoothed Particle Hydrodynamics 4.5 Lennard-Jones Molecular Dynamics 4.6 Triangulation refinement 4.7 Conway's Game of Life 4.8 Gaussian Elimination 4.9 Conclusion 5 Parallelizability of Particle Methods 5.1 Introduction 5.2 Particle Methods on Shared Memory Systems 5.2.1 Parallelization Scheme 5.2.2 Lemmata 5.2.3 Parallelizability 5.2.4 Time Complexity 5.2.5 Application 5.3 Particle Methods on Distributed Memory Systems 5.3.1 Parallelization Scheme 5.3.2 Lemmata 5.3.3 Parallelizability 5.3.4 Bounds on Time Complexity and Parallel Scalability 5.4 Conclusion 6 Turing Powerfulness and Halting Decidability 6.1 Introduction 6.2 Turing Machine 6.3 Turing Powerfulness of Particle Methods Under a First Set of Constraints 6.4 Turing Powerfulness of Particle Methods Under a Second Set of Constraints 6.5 Halting Decidability of Particle Methods 6.6 Conclusion 7 Particle Methods as a Basis for Scientific Software Engineering 7.1 Introduction 7.2 Design of the Prototype 7.3 Applications, Comparisons, Convergence Study, and Run-time Evaluations 7.4 Conclusion 8 Results, Discussion, Outlook, and Conclusion 8.1 Problem 8.2 Results 8.3 Discussion 8.4 Outlook 8.5 Conclusion

This thesis makes two noteworthy contributions in the are of mesh-free methods: a Pseudo-Divergence-Free (PDF) Element Free Galerkin (EFG) method which alleviates the volumetric locking and a Stabilized Updated Lagrangian formulation which allows to solve fast-transient dynamic problems involving large distortions. The thesis is organized in the following way. First of all, this thesis dedicates one chapter to the state of the art of mesh-free methods. The main reason is because there are many mesh-free methods that can be found in the literature which can be based on different ideas and with different properties. There is a real need of classifying, ordering and comparing these methods: in fact, the same or almost the same method can be found with different names in the literature. Secondly, a novel improved formulation of the (EFG) method is proposed in order to alleviate volumetric locking. It is based on a pseudo-divergence-free interpolation. Using the concept of diffuse derivatives an a convergence theorem of these derivatives to the ones of the exact solution, the new approximation proposed is obtained imposing a zero diffuse divergence. In this way is guaranteed that the method verifies asymptotically the incompressibility condition and in addition the imposition can be done a priori. This means that the main difference between standard EFG and the improved method is how is chosen the interpolation basis. Modal analysis and numerical results for two classical benchmark tests in solids corroborate that, as expected, diffuse derivatives converge to the derivatives of the exact solution when the discretization is refined (for a fixed dilation parameter) and, of course, that diffuse divergence converges to the exact divergence with the expected theoretical rate. For standard EFG the typical convergence rate is degrade as the incompressible limit is approached but with the improved method good results are obtained even for a nearly incompressible case and a moderately fine discretization. The improved method has also been used to solve the Stokes equations. In this case the LBB condition is not explicitly satisfied because the pseudo-divergence-free approximation is employed. Reasonable results are obtained in spite of the equal order interpolation for velocity and pressure. Finally, several techniques have been developed in the past to solve the well known tensile instability in the SPH (Smooth Particle Hydrodynamics) mesh-free method. It has been proved that a Lagrangian formulation removes completely the instability (but zero energy modes exist). In fact, Lagrangian SPH works even better than the Finite Element Method in problems involving distortions. Nevertheless, in problems with very large distortions a Lagrangian formulation will need of frequent updates of the reference configuration. When such updates are incorporated then zero energy modes are more likely to be activated. When few updates are carried out the error is small but when updates are performed frequently the solution is completely spoilt because of the zero energy modes. In this thesis an updated Lagrangian formulation is developed. It allows to carry out updates of the reference configuration without suffering the appearance of spurious modes. To update the Lagrangian formulation an incremental approach is used: an intermediate configuration will be the new reference configuration for the next time steps. It has been observed that this updated formulation suffers from similar numerical fracture to the Eulerian case. A modal analysis has proven that there exist zero energy modes. In the paper the updated Lagrangian method is exposed in detail, a stability analysis is performed and finally a stabilization technique is incorporated to preclude spurious modes.

A couple of discretization schemes, based on an FE-like tessellation of the domain and polynomial reproducing, globally smooth shape functions, are considered and numerically explored to a limited extent. The first one among these is an existing scheme, the smooth DMS-FEM, that employs Delaunay triangulation or tetrahedralization (as approximate) towards discretizing the domain geometry employs triangular (tetrahedral) B-splines as kernel functions en route to the construction of polynomial reproducing functional approximations. In order to verify the numerical accuracy of the smooth DMS-FEM vis-à-vis the conventional FEM, a Mindlin-Reissner plate bending problem is numerically solved. Thanks to the higher order continuity in the functional approximant and the consequent removal of the jump terms in the weak form across inter-triangular boundaries, the numerical accuracy via the DMS-FEM approximation is observed to be higher than that corresponding to the conventional FEM. This advantage notwithstanding, evaluations of DMS-FEM based shape functions encounter singularity issues on the triangle vertices as well as over the element edges. This shortcoming is presently overcome through a new proposal that replaces the triangular B-splines by simplex splines, constructed over polygonal domains, as the kernel functions in the polynomial reproduction scheme. Following a detailed presentation of the issues related to its computational implementation, the new method is numerically explored with the results attesting to a higher attainable numerical accuracy in comparison with the DMS-FEM.