Die Relaxation in der Variationsrechnung führt zu Minimierungsaufgaben mit einer quasi-konvexen Energiedichte. In der nichtlinearen Elastizität, Topologieoptimierung, oder bei Mehrphasenmodellen sind solche Energiedichten konvex mit einer zusätzlichen Kontrolle in der dualen Variablen und einem beidseitigem Wachstum der Ordnung $p$. Diese Minimierungsprobleme haben im Allgemeinen mehrere Lösungen, welche dennoch eine eindeutige Spannung $\sigma$ definieren. Die Approximation mit der „hybrid high-order“ (HHO) Methode benutzt eine Rekonstruktion des Gradienten in dem Raum der stückweisen Raviart-Thomas Finiten Elemente ohne Stabilisierung auf einer Triangulierung in Simplexen. Die Anwendung dieser Methode auf die Klasse der degenerierten, konvexen Minimierungsprobleme liefert eine eindeutig bestimmte, $H(\div)$ konforme Approximation $\sigma_h$ der Spannung. Die a priori Abschätzungen in dieser Arbeit gelten für gemischten Randbedingungen ohne weitere Voraussetzung an der primalen Variablen und erlauben es, Konvergenzraten bei glatten Lösungen vorherzusagen. Die a posteriori Analysis führt auf garantierte obere Fehlerschranken, eine berechenbare untere Energieschranke, sowie einen konvergenten adaptiven Algorithmus. Die numerischen Beispiele zeigen höhere Konvergenzraten mit zunehmenden Polynomgrad und bestätigen empirisch die superlineare Konvergenz der unteren Energieschranke. Obwohl der Fokus dieser Arbeit auf die nicht stabilisierte HHO Methode liegt, wird eine detaillierte Fehleranalysis für die stabilisierte Version mit einer Gradientenrekonstruktion im Raum der stückweisen Polynome präsentiert. / The relaxation procedure in the calculus of variations leads to minimization problems with a quasi-convex energy density. In some problems of nonlinear elasticity, topology optimization, and multiphase models, the energy density is convex with some convexity control plus two-sided $p$-growth. The minimizers may be non-unique in the primal variable, but define a unique stress variable $\sigma$. The approximation by hybrid high-order (HHO) methods utilizes a reconstruction of the gradients in the space of piecewise Raviart-Thomas finite element functions without stabilization on a regular triangulation into simplices. The application of the HHO methodology to this class of degenerate convex minimization problems allows for a unique $H(\div)$ conform stress approximation $\sigma_h$. The a priori estimates for the stress error $\sigma - \sigma_h$ in the Lebesgue norm are established for mixed boundary conditions without additional assumptions on the primal variable and lead to convergence rates for smooth solutions. The a posteriori analysis provides guaranteed error control, including a computable lower energy bound, and a convergent adaptive scheme. Numerical benchmarks display higher convergence rates for higher polynomial degrees and provide empirical evidence for the superlinear convergence of the lower energy bound. Although the focus is on the unstabilized HHO method, a detailed error analysis is provided for the stabilized version with a gradient reconstruction in the space of piecewise polynomials.

In computational fluid dynamics, we often need to solve complex problems with high precision and efficiency. We propose a three-pronged approach to attain this goal. First, we use the discontinuous Galerkin spectral element method (DG-SEM) for its high accuracy. Second, we use graphics processing units (GPUs) to perform our computations to exploit available parallel computing power. Third, we implement a parallel adaptive mesh refinement (AMR) algorithm to efficiently use our computing power where it is most needed. We present a GPU DG-SEM solver with AMR and dynamic load balancing for the 2D wave equation. The DG-SEM is a higher-order method that splits a domain into elements and represents the solution within these elements as a truncated series of orthogonal polynomials. This approach combines the geometric flexibility of finite-element methods with the exponential convergence of spectral methods. GPUs provide a massively parallel architecture, achieving a higher throughput than traditional CPUs. They are relatively new as a platform in the scientific community, therefore most algorithms need to be adapted to that new architecture. We perform most of our computations in parallel on multiple GPUs. AMR selectively refines elements in the domain where the error is estimated to be higher than a prescribed tolerance, via two mechanisms: p-refinement increases the polynomial order within elements, and h-refinement splits elements into several smaller ones. This provides a higher accuracy in important flow regions and increases capabilities of modeling complex flows, while saving computing power in other parts of the domain. We use the mortar element method to retain the exponential convergence of high-order methods at the non-conforming interfaces created by AMR. We implement a parallel dynamic load balancing algorithm to even out the load imbalance caused by solving problems in parallel over multiple GPUs with AMR. We implement a space-filling curve-based repartitioning algorithm which ensures good locality and small interfaces. While the intense calculations of the high order approach suit the GPU architecture, programming of the highly dynamic adaptive algorithm on GPUs is the most challenging aspect of this work. The resulting solver is tested on up to 64 GPUs on HPC platforms, where it shows good strong and weak scaling characteristics. Several example problems of increasing complexity are performed, showing a reduction in computation time of up to 3× on GPUs vs CPUs, depending on the loading of the GPUs and other user-defined choices of parameters. AMR is shown to improve computation times by an order of magnitude or more.

No description available.

[pt] A otimização topológica tem como objetivo encontrar a distribuição mais eficiente de material (ótima topologia) em uma determinada região, satisfazendo as restrições de projeto estabelecidas pelo usuário. Na abordagem tradicional atribui-se uma variável de projeto, constante, denominada densidade, para cada elemento finito da malha. Dessa forma, a qualidade da representação dos novos contornos da estrutura depende do nível de discretização da malha: quanto maior a quantidade de elementos, mais bem definida será a topologia da estrutura otimizada. No entanto, a utilização de malhas super-refinadas implica em um elevado custo computacional, principalmente na etapa de solução numérica das equações de equilíbrio pelo método dos elementos finitos. Este trabalho propõe uma nova estratégia computacional para o refinamento adaptativo local de malhas utilizando elementos finitos poligonais em domínios bidimensionais arbitrários. A ideia consiste em realizar um refinamento da malha nas regiões de concentração de material, sobretudo nos contornos internos e externos, e um desrefinamento nas regiões de baixa concentração de material, como por exemplo, nos furos internos. Desta forma, é possível obter topologias ótimas, com alta resolução e relativamente baixo custo computacional. Exemplos representativos são apresentados para demonstrar a robustez e a eficiência da metodologia proposta por meio de comparações com resultados obtidos com malhas super-refinadas e mantidas constantes durante todo o processo de otimização topológica. / [en] Topology optimization aims to find the most efficient distribution of material (optimal topology) in a given domain, subjected to design constraints defined by the user. The quality of the new boundary representation depends on the level of mesh refinement: the greater the number of elements in the mesh, the better will be the representation of the optimized structure. However, the use of super refined meshes implies in a high computational cost, especially regarding the numerical solution of the linear systems of equations that arise from the finite element method. This work proposes a new computational strategy for adaptive local mesh refinement using polygonal finite elements in arbitrary two-dimensional domains. The idea is to perform a mesh refinement in regions of material concentration, mostly in inner and outer boundaries, and a mesh derefinement in regions of low material concentration such as the internal holes. Thus, it is possible to obtain optimal topologies with high resolution and relatively low computational cost. Representative examples are presented to demonstrate the robustness and efficiency of the proposed methodology by comparing the results obtained herein with the ones from the literature where super refined meshes are held constant throughout all topology optimization process.

A simulação computacional de escoamentos incompressíveis multifásicos tem avançado continuamente e é uma área extremamente importante em Dinâmica de Fluidos Computacional (DFC) por suas várias aplicações na indústria, em medicina e em biologia, apenas para citar alguns exemplos. Apresentamos modelos matemáticos e métodos numéricos tendo em vista simulações computacionais de fluidos bifásicos newtonianos e viscoelásticos (não newtonianos), em seus regimes transiente e estacionário de escoamento. Os ingredientes principais requeridos são o Modelo de Um Fluido e o Método da Fronteira Imersa em malhas adaptativas, usados em conjunto com os métodos da Projeção de Chorin-Temam e de Uzawa. Tais metodologias são obtidas a partir de equações a derivadas parciais simples as quais, naturalmente, são resolvidas em malhas adaptativas empregando métodos multinível-multigrid. Em certas ocasiões, entretanto, para escoamentos modelados pelas equações de Navier-Stokes (e.g. em problemas onde temos altos saltos de massa específica), tem-se problemas de convergência no escopo destes métodos. Além disso, no caso de escoamentos estacionários, resolver as equações de Stokes em sua forma discreta por tais métodos não é uma tarefa fácil. Verificamos que zeros na diagonal do sistema linear resultante impedem que métodos de relaxação usuais sejam empregados. As dificuldades mencionadas acima motivaram-nos a pesquisar por, a propor e a desenvolver alternativas à metodologia multinível-multigrid. No presente trabalho, propomos métodos para obter explicitamente as matrizes que representam os sistemas lineares oriundos da discretização daquelas equações a derivadas parciais simples que são a base dos métodos de Projeção e de Uzawa. Ter em mãos estas representações matriciais é vantajoso pois com elas podemos caracterizar tais sistemas lineares em termos das propriedades de seus raios espectrais, suas definições e simetria. Muito pouco (ou nada) se sabe efetivamente sobre estes sistemas lineares associados a discretizações em malhas compostas bloco-estruturadas. É importante salientarmos que, além disso, ganhamos acesso ao uso de bibliotecas numéricas externas, como o PETSc, com seus pré-condicionadores e métodos numéricos, seriais e paralelos, para resolver sistemas lineares. Infraestrutura para nossos desenvolvimentos foi propiciada pelo código denominado ``AMR2D\'\', um código doméstico para problemas em DFC que vem sendo cuidado ao longo dos anos pelos grupos de pesquisa em DFC do IME-USP e da FEMEC-UFU. Estendemos este código, adicionando módulos para escoamentos viscoelásticos e para escoamentos estacionários modelados pelas equações de Stokes. Além disso, melhoramos de maneira notável as rotinas de cálculo de valores fantasmas. Tais melhorias permitiram a implementação do Método dos Gradientes Bi-Conjugados, baseada em visitas retalho-a-retalho e varreduras da estrutura hierárquica nível-a-nível, essencial à implementação do Método de Uzawa. / Numerical simulation of incompressible multiphase flows has continuously of advanced and is an extremely important area in Computational Fluid Dynamics (CFD) because its several applications in industry, in medicine, and in biology, just to mention a few of them. We present mathematical models and numerical methods having in sight the computational simulation of two-phase Newtonian and viscoelastic fluids (non-Newtonian fluids), in the transient and stationary flow regimes. The main ingredients required are the One-fluid Model and the Immersed Boundary Method on dynamic, adaptive meshes, in concert with Chorin-Temam Projection and the Uzawa methods. These methodologies are built from simple linear partial differential equations which, most naturally, are solved on adaptive grids employing mutilevel-multigrid methods. On certain occasions, however, for transient flows modeled by the Navier-Stokes equations (e.g. in problems where we have high density jumps), one has convergence problems within the scope of these methods. Also, in the case of stationary flows, solving the discrete Stokes equations by those methods represents no straight forward task. It turns out that zeros in the diagonal of the resulting linear systems coming from the discrete equations prevent the usual relaxation methods from being used. Those difficulties, mentioned above, motivated us to search for, to propose, and to develop alternatives to the multilevel-multigrid methodology. In the present work, we propose methods to explicitly obtain the matrices that represent the linear systems arising from the discretization of those simple linear partial differential equations which form the basis of the Projection and Uzawa methods. Possessing these matrix representations is on our advantage to perform a characterization of those linear systems in terms of their spectral, definition, and symmetry properties. Very little is known about those for adaptive mesh discretizations. We highlight also that we gain access to the use of external numerical libraries, such as PETSc, with their preconditioners and numerical methods, both in serial and parallel versions, to solve linear systems. Infrastructure for our developments was offered by the code named ``AMR2D\'\' - an in-house CFD code, nurtured through the years by IME-USP and FEMEC-UFU CFD research groups. We were able to extend that code by adding a viscoelastic and a stationary Stokes solver modules, and improving remarkably the patchwise-based algorithm for computing ghost values. Those improvements proved to be essential to allow for the implementation of a patchwise Bi-Conjugate Gradient Method which ``powers\'\' Uzawa Method.

A simulação computacional de escoamentos incompressíveis multifásicos tem avançado continuamente e é uma área extremamente importante em Dinâmica de Fluidos Computacional (DFC) por suas várias aplicações na indústria, em medicina e em biologia, apenas para citar alguns exemplos. Apresentamos modelos matemáticos e métodos numéricos tendo em vista simulações computacionais de fluidos bifásicos newtonianos e viscoelásticos (não newtonianos), em seus regimes transiente e estacionário de escoamento. Os ingredientes principais requeridos são o Modelo de Um Fluido e o Método da Fronteira Imersa em malhas adaptativas, usados em conjunto com os métodos da Projeção de Chorin-Temam e de Uzawa. Tais metodologias são obtidas a partir de equações a derivadas parciais simples as quais, naturalmente, são resolvidas em malhas adaptativas empregando métodos multinível-multigrid. Em certas ocasiões, entretanto, para escoamentos modelados pelas equações de Navier-Stokes (e.g. em problemas onde temos altos saltos de massa específica), tem-se problemas de convergência no escopo destes métodos. Além disso, no caso de escoamentos estacionários, resolver as equações de Stokes em sua forma discreta por tais métodos não é uma tarefa fácil. Verificamos que zeros na diagonal do sistema linear resultante impedem que métodos de relaxação usuais sejam empregados. As dificuldades mencionadas acima motivaram-nos a pesquisar por, a propor e a desenvolver alternativas à metodologia multinível-multigrid. No presente trabalho, propomos métodos para obter explicitamente as matrizes que representam os sistemas lineares oriundos da discretização daquelas equações a derivadas parciais simples que são a base dos métodos de Projeção e de Uzawa. Ter em mãos estas representações matriciais é vantajoso pois com elas podemos caracterizar tais sistemas lineares em termos das propriedades de seus raios espectrais, suas definições e simetria. Muito pouco (ou nada) se sabe efetivamente sobre estes sistemas lineares associados a discretizações em malhas compostas bloco-estruturadas. É importante salientarmos que, além disso, ganhamos acesso ao uso de bibliotecas numéricas externas, como o PETSc, com seus pré-condicionadores e métodos numéricos, seriais e paralelos, para resolver sistemas lineares. Infraestrutura para nossos desenvolvimentos foi propiciada pelo código denominado ``AMR2D\'\', um código doméstico para problemas em DFC que vem sendo cuidado ao longo dos anos pelos grupos de pesquisa em DFC do IME-USP e da FEMEC-UFU. Estendemos este código, adicionando módulos para escoamentos viscoelásticos e para escoamentos estacionários modelados pelas equações de Stokes. Além disso, melhoramos de maneira notável as rotinas de cálculo de valores fantasmas. Tais melhorias permitiram a implementação do Método dos Gradientes Bi-Conjugados, baseada em visitas retalho-a-retalho e varreduras da estrutura hierárquica nível-a-nível, essencial à implementação do Método de Uzawa. / Numerical simulation of incompressible multiphase flows has continuously of advanced and is an extremely important area in Computational Fluid Dynamics (CFD) because its several applications in industry, in medicine, and in biology, just to mention a few of them. We present mathematical models and numerical methods having in sight the computational simulation of two-phase Newtonian and viscoelastic fluids (non-Newtonian fluids), in the transient and stationary flow regimes. The main ingredients required are the One-fluid Model and the Immersed Boundary Method on dynamic, adaptive meshes, in concert with Chorin-Temam Projection and the Uzawa methods. These methodologies are built from simple linear partial differential equations which, most naturally, are solved on adaptive grids employing mutilevel-multigrid methods. On certain occasions, however, for transient flows modeled by the Navier-Stokes equations (e.g. in problems where we have high density jumps), one has convergence problems within the scope of these methods. Also, in the case of stationary flows, solving the discrete Stokes equations by those methods represents no straight forward task. It turns out that zeros in the diagonal of the resulting linear systems coming from the discrete equations prevent the usual relaxation methods from being used. Those difficulties, mentioned above, motivated us to search for, to propose, and to develop alternatives to the multilevel-multigrid methodology. In the present work, we propose methods to explicitly obtain the matrices that represent the linear systems arising from the discretization of those simple linear partial differential equations which form the basis of the Projection and Uzawa methods. Possessing these matrix representations is on our advantage to perform a characterization of those linear systems in terms of their spectral, definition, and symmetry properties. Very little is known about those for adaptive mesh discretizations. We highlight also that we gain access to the use of external numerical libraries, such as PETSc, with their preconditioners and numerical methods, both in serial and parallel versions, to solve linear systems. Infrastructure for our developments was offered by the code named ``AMR2D\'\' - an in-house CFD code, nurtured through the years by IME-USP and FEMEC-UFU CFD research groups. We were able to extend that code by adding a viscoelastic and a stationary Stokes solver modules, and improving remarkably the patchwise-based algorithm for computing ghost values. Those improvements proved to be essential to allow for the implementation of a patchwise Bi-Conjugate Gradient Method which ``powers\'\' Uzawa Method.

Este é o primeiro trabalho que apresenta simulações tridimensionais completamente adaptativas de um modelo de campo de fase para um fluido incompressível com densidade de massa constante e viscosidade variável, conhecido como Modelo H. Solucionando numericamente as equações desse modelo em malhas refinadas localmente com a técnica AMR, simulamos computacionalmente escoamentos bifásicos tridimensionais. Os modelos de campo de fase oferecem uma aproximação física sistemática para investigar fenômenos que envolvem sistemas multifásicos complexos, tais como fluidos com camadas de mistura, a separação de fases sob forças de cisalhamento e a evolução de micro-estruturas durante processos de solidificação. Como as interfaces são substituídas por delgadas regiões de transição (interfaces difusivas), as simulações de campo de fase requerem muita resolução nessas regiões para capturar corretamente a física do problema em estudo. Porém essa não é uma tarefa fácil de ser executada numericamente. As equações que caracterizam o modelo de campo de fase contêm derivadas de ordem elevada e intrincados termos não lineares, o que exige uma estratégia numérica eficiente capaz de fornecer precisão tanto no tempo quanto no espaço, especialmente em três dimensões. Para obter a resolução exigida no tempo, usamos uma discretização semi-implícita de segunda ordem para solucionar as equações acopladas de Cahn-Hilliard e Navier-Stokes (Modelo H). Para resolver adequadamente as escalas físicas relevantes no espaço, utilizamos malhas refinadas localmente que se adaptam dinamicamente para recobrir as regiões de interesse do escoamento, como por exemplo, as vizinhanças das interfaces do fluido. Demonstramos a eficiência e a robustez de nossa metodologia com simulações que incluem a separação dos componentes de uma mistura bifásica, a deformação de gotas sob cisalhamento e as instabilidades de Kelvin-Helmholtz. / This is the first work that introduces 3D fully adaptive simulations for a phase field model of an incompressible fluid with matched densities and variable viscosity, known as Model H. Solving numerically the equations of this model in meshes locally refined with AMR technique, we simulate computationally tridimensional two-phase flows. Phase field models offer a systematic physical approach to investigate complex multiphase systems phenomena such as fluid mixing layers, phase separation under shear and microstructure evolution during solidification processes. As interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations need great resolution in these regions to capture correctly the physics of the studied problem. However, this is not an easy task to do numerically. Phase field model equations have high order derivatives and intricate nonlinear terms, which require an efficient numerical strategy that can achieve accuracy both in time and in space, especially in three dimensions. To obtain the required resolution in time, we employ a semi-implicit second order discretization scheme to solve the coupled Cahn-Hilliard/Navier-Stokes equations (Model H). To resolve adequatly the relevant physical scales in space, we use locally refined meshes which adapt dynamically to cover special flow regions, e.g., the vicinity of the fluid interfaces. We demonstrate the efficiency and robustness of our methodology with simulations that include spinodal decomposition, the deformation of drops under shear and Kelvin-Helmholtz instabilities.

本文目的在運用Adaptive Mesh Model，以具有高解析度的細網結構(fine mesh)來評價重設選擇權，以解決傳統Ritchken(1995) 樹狀模型在運用上會出現一些無法有效率運算和收斂狀況不佳二個問題。 本文的貢獻在發現評價重設選擇權可以使用下降生效界限選擇權的原理來探討。另外為解決上述Ritchken模型所面臨的二個問題，應用Adaptive Mesh Model於下生效界限選擇權，設置適當之細網結構，演算出更精確的重設選擇權價格。而且本樹網模型不再受限於美式下出界選擇權與美式下生效界限選擇權的組合不一定是美式重設選擇權的困擾。故在納入Adaptive Mesh Model的下降生效界限選擇權來評價歐式或美式重設選擇權，可以獲得良好的收斂效率。

Ideal magnetohydrodynamics (MHD) simulations are known to have problems in satisfying the solenoidal constraint (i.e. the divergence of magnetic field should be equal to zero, $ ablacdotvec{B} = 0$). The simulations become unstable unless specific measures have been taken. In this thesis, a solenoidal constraint satisfying technique that allows discrete satisfaction of the solenoidal constraint up to the machine accuracy is presented and validated with a variety of test cases. Due to its inspiration from Chorin's artificial compressibility method developed for incompressible CFD applications, the technique was named as extit{artificial compressibility analogy (ACA)} approach. It is demonstrated that ACA is a purely hyperbolic, stable and consistent technique, which is moreover easy to implement. Unlike some other techniques, it does not pose any problems of the sort that $ ablacdotvec{B}$ errors accumulate in the vicinity of the stagnant regions of flow. With these crucial properties, ACA is thought to be a remedy to the drawbacks of the most commonly used solenoidal constraint satisfying techniques in the literature namely: Incorrect shock capturing and poor performance of the convective stabilization mechanism in regions of stagnant flow for Powell's source term method; exceedingly complex implementation for constrained transport technique due to the staggered grid representation; computationally expensive nature due to the necessity of a Poisson solver combined with hyperbolic/elliptic numerical methods for classical projection schemes. In the first chapter of the thesis, general background knowledge is given about plasmas, MHD and its history, a certain class of upwind finite volume methods, namely Riemann solvers, and their applications in MHD, the definition, constituents, formation mechanisms and effects of space weather and some of the space missions that are and will be performed in its prediction. Secondly, detailed analysis of the compressible ideal MHD equations is given in the form of the derivation of the equations, their dimensionless numbers which will be of use to specify the flows in the following chapters, and finally, the presentation of the MHD waves and discontinuities, which indicates the complexity of the system of ideal MHD equations and therefore their further numerical analysis. The next discussion is about the main subject of the thesis, namely the solenoidal constraint satisfying techniques. First of all, the definition and significance of the solenoidal constraint is given. Afterwards, the most common solenoidal constraint satisfying techniques in the literature are reviewed along with their abovementioned drawbacks. Moreover, particular emphasis is given to the Powell's source term approach which was also implemented in the upwind finite volume MHD solver developed. In addition, the hyperbolic divergence cleaning technique is presented in detail together with the resemblance and differences between it and ACA. Some other solenoidal constraint satisfying techniques are briefly mentioned at this stage. After these, ACA is presented in the following way: The point of inspiration, which is the analogy made with Chorin's artificial compressibility method developed for incompressible CFD, the introduction of the modified system of ideal MHD equations due to ACA, the derivation of the wave equation governing the propagation of $ ablacdotvec{B}$ errors and the analytical consistency proof. Having finished the core discussion of the thesis, the solver developed and its constituents are given in the fourth chapter. Furthermore, a brief overview of the platform into which this solver was implemented, namely COOLFluiD, is also given at this point. Afterwards, a thorough numerical verification of the ACA approach has been made on an increasingly complex suite of test cases. The results obtained with ACA and Powell's source term implementations are given in order to numerically analyse and verify ACA and compare the two methods and validate them with the results from literature. The sixth chapter is devoted to further validation of ACA performed with a variety of more advanced space weather-related simulations. In this chapter, also the $vec{B}_{ extrm{0}} + vec{B}_{ extrm{1}}$ splitting technique used to treat planetary magnetosphere is presented along with its application to ACA and Powell's source term approaches. This technique is utilized in obtaining the solar wind/Earth's magnetosphere interaction results and is based on suppressing the direct inclusion of the Earth's magnetic field, which is a dipole field, in the solution variables. In this way, problems are avoided with the energy equation that could arise from the drastic change of the ratio of the dipole field and the variable field computed by the solver (i.e. $frac{lvertvec{B}_{ extrm{0}}lvert}{lvertvec{B}_{ extrm{1}}lvert}$) in the computational domain. Finally, conclusions and future perspectives related to the material presented in the thesis are put forward.

In this dissertation, we develop Adaptive Mesh Refinement (AMR) techniques for the steady-state multigroup SN neutron transport equation using a higher-order Discontinuous Galerkin Finite Element Method (DGFEM). We propose two error estimations, a projection-based estimator and a jump-based indicator, both of which are shown to reliably drive the spatial discretization error down using h-type AMR. Algorithms to treat the mesh irregularity resulting from the local refinement are implemented in a matrix-free fashion. The DGFEM spatial discretization scheme employed in this research allows the easy use of adapted meshes and can, therefore, follow the physics tightly by generating group-dependent adapted meshes. Indeed, the spatial discretization error is controlled with AMR for the entire multigroup SNtransport simulation, resulting in group-dependent AMR meshes. The computing efforts, both in memory and CPU-time, are significantly reduced. While the convergence rates obtained using uniform mesh refinement are limited by the singularity index of transport solution (3/2 when the solution is continuous, 1/2 when it is discontinuous), the convergence rates achieved with mesh adaptivity are superior. The accuracy in the AMR solution reaches a level where the solution angular error (or ray effects) are highlighted by the mesh adaptivity process. The superiority of higherorder calculations based on a matrix-free scheme is verified on modern computing architectures. A stable symmetric positive definite Diffusion Synthetic Acceleration (DSA) scheme is devised for the DGFEM-discretized transport equation using a variational argument. The Modified Interior Penalty (MIP) diffusion form used to accelerate the SN transport solves has been obtained directly from the DGFEM variational form of the SN equations. This MIP form is stable and compatible with AMR meshes. Because this MIP form is based on a DGFEM formulation as well, it avoids the costly continuity requirements of continuous finite elements. It has been used as a preconditioner for both the standard source iteration and the GMRes solution technique employed when solving the transport equation. The variational argument used in devising transport acceleration schemes is a powerful tool for obtaining transportconforming diffusion schemes. xuthus, a 2-D AMR transport code implementing these findings, has been developed for unstructured triangular meshes.