In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error. The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution. The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form. Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.

In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error. The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution. The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form. Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.

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Many physical problems lead to boundary value problems for partial differential equations, which can be solved with the finite element method. In order to construct adaptive solution algorithms or to measure the error one aims at reliable a posteriori error estimators. Many such estimators are known, as well as their theoretical foundation. Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet. For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility. For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L_2 error estimator, respectively. A corresponding mathematical theory is given.For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes. The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role. AMS(MOS): 65N30, 65N15, 35B25

This thesis is a blend of neutron transport theory and numerical analysis. We start with the study of the problem of the Mika/Case eigenexpansion used in the solution process of the homogeneous one-speed Boltzmann neutron transport equation with anisotropic scattering for plane symmetry. The anisotropic scattering is expressed as a finite Legendre series in which the coefficients are the ``scattering coefficients'. This eigenexpansion consists of a discrete spectrum of eigenvalues with its corresponding eigenfunctions and the continuous spectrum [-1,+1] with its corresponding eigendistributions. In the general case where the anisotropic scattering can be of any (finite) order, multiple discrete eigenvalues exist and these have to be located to have the complete spectrum. We have devised a stable and robust method that locates all these discrete eigenvalues. The method is a two-step process: first the number of discrete eigenvalues is calculated and this is followed by the calculation of the discrete eigenvalues themselves, now being able to count them down and make sure none are forgotten. During our numerical experiments, we came across what we called near-singular eigenvalues: discrete eigenvalues that are located extremely close to the continuum and hence lead to near-singular behaviour in the eigenfunction. Our solution method has been adapted and allows for the automatic detection of such a near-singular eigenvalue. For the elements of the continuous spectrum [-1,+1], there is no non-zero function satisfying the associated eigenequation but there is a non-zero distribution that does satisfy it. It is not feasible to compute a distribution as such but one can evaluate integrals in which this distribution appears. The continuum part of the eigenexpansion can hence only be characterised by its (angular) moments. Accurate and fast numerical quadrature is needed to evaluate these integrals. Several quadrature methods have been evaluated on a representative test function. The eigenexpansion was proved to be orthogonal and complete and hence can be used to represent the infinite medium Green's function. The latter is the building block of the Boundary Sources Method, an integral solution method for the neutron transport equation. Using angular and angular/spatial moments of the Green's function, it is possible to solve with high accuracy slab problems. We have written a one-dimensional slab code implementing this Boundary Sources Method allowing for media with arbitrary order anisotropic scattering. Our results are very good and the code can be considered as a benchmark code for others. As a final application, we have used our code to study the discrete spectrum of a well-known scattering kernel in radiative transfer, the Henyey-Greenstein kernel. This kernel has one free parameter which is used to fit the kernel to experimental data. Since the kernel is a continuous function, a finite Legendre approximation needs to be adopted. Depending on the free parameter, the approximation order and the number of secondaries per collision, the number of discrete eigenvalues ranges from two to thirty and even more. Bounds for the minimum approximation order are derived for different requirements on the approximation: non-negativity, an absolute and relative error tolerance.

<p>Domain decomposition method provides a solution for the very large electromagnetic</p><p>system which are impossible for single domain methods. Discontinuous Galerkin</p><p>(DG) method can be viewed as an extreme version of the domain decomposition,</p><p>i.e., each element is regarded as one subdomain. The whole system is solved element</p><p>by element, thus the inversion of the large global system matrix is no longer necessary,</p><p>and much larger system can be solved with the DG method compared to the</p><p>continuous Galerkin (CG) method.</p><p>In this work, the DG method is implemented on a subdomain level, that is, each subdomain contains multiple elements. The numerical flux only applies on the</p><p>interfaces between adjacent subdomains. The subodmain level DG method divides</p><p>the original large global system into a few smaller ones, which are easier to solve,</p><p>and it also provides the possibility of parallelization. Compared to the conventional</p><p>element level DG method, the subdomain level DG has the advantage of less total</p><p>DoFs and fexibility in interface choice. In addition, the implicit time stepping is </p><p>relatively much easier for the subdomain level DG, and the total CPU time can be</p><p>much less for the electrically small or multiscale problems.</p><p>The hybrid of elements are employed to reduce the total DoF of the system.</p><p>Low-order tetrahedrons are used to catch the geometry ne parts and high-order</p><p>hexahedrons are used to discretize the homogeneous and/or geometry coarse parts.</p><p>In addition, the non-conformal mesh not only allow dierent kinds of elements but</p><p>also sharp change of the element size, therefore the DoF can be further decreased.</p><p>The DGTD method in this research is based on the EB scheme to replace the</p><p>previous EH scheme. Dierent from the requirement of mixed order basis functions</p><p>for the led variables E and H in the EH scheme, the EB scheme can suppress the</p><p>spurious modes with same order of basis functions for E and B. One order lower in</p><p>the basis functions in B brings great benets because the DoFs can be signicantly</p><p>reduced, especially for the tetrahedrons parts.</p><p>With the basis functions for both E and B, the EB scheme upwind </p><p>ux and</p><p>EB scheme Maxwellian PML, the eigen-analysis and numerical results shows the</p><p>eectiveness of the proposed DGTD method, and multiscale problems are solved</p><p>eciently combined with the implicit-explicit hybrid time stepping scheme and multiple</p><p>kinds of elements.</p><p>The EB scheme DGTD method is further developed to allow arbitrary anisotropic</p><p>media via new anisotropic EB scheme upwind </p><p>ux and anisotropic EB scheme</p><p>Maxwellian PML. The anisotropic M-PML is long time stable and absorb the outgoing</p><p>wave eectively. A new TF/SF boundary condition is brought forward to</p><p>simulate the half space case. The negative refraction in YVO4 bicrystal is simulated</p><p>with the anisotropic DGTD and half space TF/SF condition for the rst time with</p><p>numerical methods.</p> / Dissertation

Acoustic source localization (ASL) is pinpointing an acoustic source. ASL can reveal the point of impact of a foreign object or the point of crack initiation in a structure. ASL is necessary for continuous health monitoring of a structure. ASL in an anisotropic plate is a challenging task. This dissertation aims to investigate techniques that are currently being used to precisely determine an acoustic source location in an anisotropic plate without knowing its material properties. A new technique is developed and presented here to overcome the existing shortcomings of the acoustic source localization in anisotropic plates. It is done by changing the analysis perspective from the angular dependent group velocity of the wave and its straight line propagation to the wave front shapes and their geometric properties when a non-circular wave front is generated. Especially, 'rhombic wave front' and 'elliptical wave front' are dealt with because they are readily observed in highly anisotropic composite plates. Once each proposed technique meets the requirements of measurement, four sensor clusters in three different quadrants (recorded by 12 sensors) for the rhombus and at least three sensor clusters (recorded by 9 sensors) for the ellipse, accurate Acoustic Source Localization is obtained. It has been successfully demonstrated in the numerical simulations. In addition, a series of experimental tests demonstrate reliable and robust prediction performance of the developed new acoustic source localization technique.

Many solids have Fermi surfaces which are approximated as ellipsoids. A comprehensive solution for the magnetoconductivity of an ellipsoid is obtained which proves the existence of a relaxation time tensor which can be anisotropic and which is a function of energy only.

A partir de considerações sobre interações entre pares de objetos dependentes apenas de suas orientações relativas, desenvolvemos um modelo discreto para cristais líquidos biaxiais. Representamos estes sistemas por um conjunto de partículas biaxiais localizadas nos sítios de uma rede cristalina e que se orientam somente ao longo dos três eixos cartesianos. O diagrama de fases, obtido na aproximação de campo médio, apresenta uma fase nemática biaxial, duas fases nemáticas uniaxiais, além da fase isotrópica. Estas fases se encontram em um ponto multicrítico em cuja vizinhança fazemos a expansão de Landau da Energia Livre. Finalmente analisamos a adição de um campo externo favorável ao ordenamento nemático uniaxial. / From considerations about the pair interaction of objects depending only on their relative orientations, we develop a discrete model for biaxial liquid crystals. These systems are represented by a set of biaxial particles localized on the sites of a crystal lattice and oriented only along the three Cartesian axes. The phase diagram, obtained in a mean field approximation, displays a biaxial nematic phase, two uniaxial nematic phases, beside the isotropic phase. These phases meet on a multicritical point about which we make the Landau expansion of the free energy. Finally, we analyse the case where an external field is applied to favour the uniaxial nematic ordering.

A partir de considerações sobre interações entre pares de objetos dependentes apenas de suas orientações relativas, desenvolvemos um modelo discreto para cristais líquidos biaxiais. Representamos estes sistemas por um conjunto de partículas biaxiais localizadas nos sítios de uma rede cristalina e que se orientam somente ao longo dos três eixos cartesianos. O diagrama de fases, obtido na aproximação de campo médio, apresenta uma fase nemática biaxial, duas fases nemáticas uniaxiais, além da fase isotrópica. Estas fases se encontram em um ponto multicrítico em cuja vizinhança fazemos a expansão de Landau da Energia Livre. Finalmente analisamos a adição de um campo externo favorável ao ordenamento nemático uniaxial. / From considerations about the pair interaction of objects depending only on their relative orientations, we develop a discrete model for biaxial liquid crystals. These systems are represented by a set of biaxial particles localized on the sites of a crystal lattice and oriented only along the three Cartesian axes. The phase diagram, obtained in a mean field approximation, displays a biaxial nematic phase, two uniaxial nematic phases, beside the isotropic phase. These phases meet on a multicritical point about which we make the Landau expansion of the free energy. Finally, we analyse the case where an external field is applied to favour the uniaxial nematic ordering.