In this thesis, we shall give a concise account for the classification of the structure of positive radial solutions of the semilinear elliptic equation$$Delta u+K(|x|)u^{p}=0 .$$ It is known that a radial solution $u$ is crossing if $u$ has a zero in $(0, infty)$; $u$ is slowly decaying if $u$ is positive but $displaystylelim_{r ightarrow{infty}}r^{n-2}u=infty$; u is rapidly decaying if $u$ is positive, $displaystylelim_{r ightarrow{infty}}r^{n-2}u$ exists and is positive. Using some Pohozaev identities, we show that under certain condition on $K$, by comparing some parameters $r_{G}$ and $r_{H}$, the structure of positive radial solutions for various initial conditions can be classified as Type Z ($u(r; alpha)$ is crossing for all $r>0$ ), Type S ($u(r; alpha)$ is slowly decaying for all $r>0$), and Type M (there is some $alpha_{f}$ such that $u(r; alpha)$ is crossing for $alphain(alpha_{f}, infty)$, $u(r; alpha)$ is slowly decaying for $alpha=alpha_{f}$, and $u(r; alpha)$ is rapidly decaying for $alphain(0, alpha_{f})$). The above work is due to Yanagida and Yotsutani.

The elliptic equation $Delta u+K(|x|)|u|^{p-1}u=0,xin mathbf{R}^{n}$ is studied, where $p>1$, $n>2$, $K(r)$ is smooth and positive on $(0,infty)$, and $rK(r)in L^{1}(0,1)$. It is known that the radial solution either oscillates infinitely, or $lim_{r ightarrow infty}r^{n-2}u(r;al) in Rsetminus {0}$ (rapidly decaying), or $lim_{r ightarrow infty}r^{n-2}u(r;al) = infty (or -infty)$ (slowly decaying). Let $u=u(r;al)$ is a solution satisfying $u(0)=al$. In this thesis, we classify all the radial solutions into three types: Type R($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is rapidly decaying at $r=infty$. Type S($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is slowly decaying at $r=infty$. Type O: $u$ has infinitely many zeros on $(0,infty)$. If $rK_{r}(r)/K(r)$ satisfies some conditions, then the structure of radial solutions is determined completely. In particular, there exists $0<al_{0}<al_{1}<al_{2}<cdots<infty$ such that $u(r;al_{i})$ is of Type R($i$), and $u(r;al)$ is of Type S($i$) for all $al in (al_{i-1},al_{i})$, where $al_{-1}:=0$. These works are due to Yanagida and Yotsutani. Their main tools are Kelvin transformation, Pr"{u}fer transformation, and a Pohozaev identity. Here we give a concise account. Also, I impose a concept so called $r-mu graph$, and give two proofs of the Pohozaev identity.

The bending of an orthotropic cusped plate in energetic and weighted Sobolev spaces has been considered. The existence and uniqueness of generalized and weak solutions of admissible boundary value problems (BVPs) have been investigated.

In the thesis, we propose a newly-developed method called the method of Connected Local Fields (CLF) to analyze opto-electromagnetic passive devices. The method of CLF somewhat resembles a hybrid between the finite difference and pseudo-spectral methods. For opto-electromagnetic passive devices, our primary concern is their steady state behavior, or narrow-band characteristics, so we use a frequency-domain method, in which the system is governed by the Helmholtz equation. The essence of CLF is to use the intrinsic general solution of the Helmholtz equation to expand the local fields on the compact stencil. The original equation can then be transformed into the discretized form called LFE-9 (in 2-D case), and the intrinsic reconstruction formulae describing each overlapping local region can be obtained. Further, we present rigorous analysis of the numerical dispersion equation of LFE-9, by means of first-order approximation, and acquire the closed-form formula of the relative numerical dispersion error. We are thereby able to grasp the tangible influences brought both by the sampling density as well as the propagation direction of plane wave on dispersion error. In our dispersion analysis, we find that the LFE-9 formulation achieves the sixth-order accuracy: the theoretical highest order for discretizing elliptic partial differential equations on a compact nine-point stencil. Additionally, the relative dispersion error of LFE-9 is less than 1%, given that sampling density greater than 2.1 points per wavelength. At this point, the sampling density is nearing that of the Nyquist-Shannon sampling limit, and therefore computational efforts can be significantly reduced.

The stress and fatigue analysis of roller chain under varied loading is investigated in this study. With the dynamic responses of a gear-cam intermittent mechanism, boundary conditions are set up to the finite element model. The commercial MARC finite element method package is used in this work. Stress distribution is our concern. ASME elliptic equation and AGMA allowable contact stress criteria are employed in the fatigue analysis. Results indicate that the maximum of von Mises stress occurs on the contact surface between pins and plates. Besides, the initial failures on this loading are predicted to happen on the plates according to the fatigue analysis.

The Lagrange multiplier used for the Dirichlet condition is well known in mathematics community, and the Lagrange multiplier used for the Neumann condition is popular for the Trefftz method in engineering community, in particular for elasticity problems. The latter is called the Hybrid Trefftz method (HTM). However, it seems to export no analysis for HTM. This paper is devoted to error analysis of the HTM for −£Gu + cu = 0 with c = 1 or c = 0. Error bounds are derived to provide the optimal convergence rates. Numerical experiments and comparisons between two kinds of Lagrange multipliers are also reported. The analysis in this paper can also be extended to the HTM for elasticity problems.

The Lagrange multiplier used for the displacement (i.e., Dirichlet) condition is well known in mathematics community (see [1, 2, 10, 18]), and the Lagrange multiplier used for the traction (i.e., Neumann)condition is popular for the Trefftz method for elasticity problems in engineering community, which is called the Hybrid Trefftz method (HTM). However, it seems to export no analysis for HTM. This paper is devoted to error analysis of the HTM for elasticity problems. Numerical experiments are reported to support the analysis made.

Cette thèse est consacrée principalement à l’étude de quelques problèmes elliptiques singuliers dans un domaine Ωɛ*, périodiquement perforé par des trous de taille ɛ. On montre l’existence et l’unicité d’une solution, pour tout ɛ fixé, ainsi que des résultats d’homogénéisation et correcteurs pour le problème singulier suivant :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Où l’on prescrit des conditions de Dirichlet homogènes sur la frontière extérieure Γɛ0 et des conditions de Robin non linéaires sur la frontière des trous Γɛ1. Le champ matriciel quasi linéaire A est elliptique, borné, périodique dans la primière variable et de Carathéodory. Le terme singulier non linéaire est le produit d’une fonction continue ζ (singulier en zéro) et de f, dont la sommabilité dépend de la croissance de ζ près de sa singularité. Le terme de bord non linéaire h est une fonction croissante de classe C1, ρ et g sont des fonctions périodiques non négatives avec sommabilité convenables. Pour étudier le comportement asymptotique du problème quand ɛ -> 0, on applique la méthode de l’éclatement périodique due à D. Cioranescu-A. Damlamian-G. Griso (cf. D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki pour les domaines perforés). Enfin, on montre l’existence et l’unicité de la solution faible pour la même équation, dans un domaine à deux composants Ω = Ω1 υ Ω2 υ Γ, étant Γ l’interface entre le composant connecté Ω1 et les inclusions Ω2. Plus précisément on considère{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Où λ est un réel non négatif et h représente le coefficient de proportionnalité entre le flux de chaleur et le saut de la solution, et il est supposé être borné et non négatif sur Γ. / This thesis is mainly devoted to the study of some singular elliptic problems posed in perforated domains. Denoting by Ωɛ* e domain perforated by ɛ-periodic holes of ɛ-size, we prove existence and uniqueness of the solution , for fixed ɛ, as well as homogenization and correctors results for the following singular problem :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Where homogeneous Dirichlet and nonlinear Robin conditions are prescribed on the exterior boundary Γɛ0 and on the boundary of the holles Γɛ1, respectively. The quasilinear matrix field A is elliptic, bounded, periodic in the first variable and Carathéodory. The nonlinear singular lower order ter mis the product of a continuous function ζ (singular in zero) and f whose summability depends on the growth of ζ near its singularity. The nonlinear boundary term h is a C1 increasing function, ρ and g are periodic nonnegative functions with prescribed summabilities. To investigate the asymptotic behaviour of the problem, as ɛ -> 0, we apply the Periodic Unfolding Method by D. Cioranescu-A. Damlamian-G. Griso, adapted to perforated domains by D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki. Finally, we show existence and uniqueness of a weak solution of the same equation in a two-component domain Ω = Ω1 υ Ω2 υ Γ, being Γ the interface between the connected component Ω1 and the inclusions Ω2. More precisely we consider{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Where ν1 is the unit external vector to Ω1 and λ a nonnegative real number. Here h represents the proportionality coefficient between the continuous heat flux and the jump of the solution and it is assumed to be bounded and nonnegative on Γ.

This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers. The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q  + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.      In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities. / <p>QC 20130926</p>

Bayesian parameter estimation is a popular method to address inverse problems. However, since prior distributions are chosen based on expert judgement, the method can inherently introduce bias into the understanding of the parameters. This can be especially relevant in the case of distributed parameters where it is difficult to check for error. To minimize this bias, we develop the idea of a minimally corrective, approximately recovering prior (MCAR prior) that generates a guide for the prior and corrects the expert supplied prior according to that guide. We demonstrate this approach for the 1D elliptic equation or the elliptic partial differential equation and observe how this method works in cases with significant and without any expert bias. In the case of significant expert bias, the method substantially reduces the bias and, in the case with no expert bias, the method only introduces minor errors. The cost of introducing these small errors for good judgement is worth the benefit of correcting major errors in bad judgement. This is particularly true when the prior is only determined using a heuristic or an assumed distribution. / Master of Science