Midwest Numerical Analysis Day 2016 Contributed Talks Theoretical and numerical approximations of the singularly perturbed convection-diffusion equations

We explore singularly perturbed convection-diffusion equations in a circular domain. Considering boundary layer analysis of the singularly perturbed equations and we show convergence results. In view of numerical analysis, We discuss approximation schemes, error estimates and numerical computations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh. Spectrum Slicing by Polynomial and Rational Function Filtering Yuanzhe Xi, University of Minnesota Abstract: Two filtering techniques are presented for solving large eigenvalue problems by spectrum slicing. In the first approach, the filter is constructed as the least-squares approximation to an appropriately centered Dirac distribution. In the second approach, a least-squares rational filter is designed for matrices whose spectrum is contained in a large interval and generalized eigenvalue problems. Two filtering techniques are presented for solving large eigenvalue problems by spectrum slicing. In the first approach, the filter is constructed as the least-squares approximation to an appropriately centered Dirac distribution. In the second approach, a least-squares rational filter is designed for matrices whose spectrum is contained in a large interval and generalized eigenvalue problems. Domain Decomposition Methods for Symmetric Eigenvalue Problems Vasileios Kalantzis, University of Minnesota Abstract: In this talk we will discuss Domain-Decomposition (DD) type methods for large Hermitian eigenvalue problems. This class of techniques rely on spectral Schur complements combined with Newtons iteration. The eigenvalues of the spectral Schur complement appear in the form of branches of some functions, the roots of which are eigenvalues of the original matrix. It is possible to extract these roots by a number of methods which range from a form or approximate Rayleigh iteration to an approximate inverse iteration, in which a Domain Decomposition framework is used. Numerical experiments in parallel environments will illustrate the numerical properties and efficiency of the method. In this talk we will discuss Domain-Decomposition (DD) type methods for large Hermitian eigenvalue problems. This class of techniques rely on spectral Schur complements combined with Newtons iteration. The eigenvalues of the spectral Schur complement appear in the form of branches of some functions, the roots of which are eigenvalues of the original matrix. It is possible to extract these roots by a number of methods which range from a form or approximate Rayleigh iteration to an approximate inverse iteration, in which a Domain Decomposition framework is used. Numerical experiments in parallel environments will illustrate the numerical properties and efficiency of the method.