In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the...
The present thesis examines the role of Hermitian Yang-Mills (HYM) connections in Spin(7) instanton moduli spaces over asymptotically conical (AC) Calabi-Yau (CY) fourfolds. The starting point is Lewis' energy estimate: Lewis: if a principal G-bundle P over a closed CY fourfold X8} admits HYM soluti...
This thesis focuses on the construction of finite element numerical homogenization schemes for both linear and selected fully-nonlinear elliptic partial differential equations in nondivergence-form. In the first part of the thesis, we study periodic homogenization problems of the form A(x/ε):D2 uε =...
The convergence rate of domain decomposition methods is generally determined by the eigenvalues of the preconditioned system. For second-order elliptic partial differential equations, coefficient discontinuities with a large contrast can lead to a deterioration of the convergence rate. Only by imple...
Spectral methods can solve elliptic partial differential equations (PDEs) numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time-dependent problems, almost all focus has been on low-order finite difference schemes for the...
We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type. Our first existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic. In this context, our result requires only a partial C2 decay of t...
This thesis is concerned with the development and analysis of a discrete counterpart of the well-known De-Giorgi-type regularity theory for solutions of elliptic partial differential equations in the setting of finite element approximations. We consider a finite element space consisting of piecewise...
Henri Berestycki, directeur d’étudesÉquations de réaction-diffusion et dynamiques de populations biologiquesDans un premier temps, les séances du séminaire ont été consacrées à l’étude générale des équations aux dérivées partielles elliptiques et paraboliques. Les principes du maximum ainsi que les...
Henri Berestycki, directeur d’étudesÉquations de réaction-diffusion et dynamiques de populations biologiquesDans un premier temps, les séances du séminaire ont été consacrées à l’étude générale des équations aux dérivées partielles elliptiques et paraboliques. Les principes du maximum ainsi que les...
This thesis concerns the analytical and practical aspects of applying the Closest Point Method to solve elliptic partial differential equations (PDEs) on smooth surfaces and domains with smooth boundaries. A new numerical scheme is proposed to solve surface elliptic PDEs and a novel geometric multig...
In this thesis numerical methods for solving elliptic partial differential equations are developed. These differential equations are discretized using finite differences and the resulting algebraic equations are then solved using iterative techniques. In particular, dynamic A.D.I. and multigrid meth...