Title 1: On the convergence of a two-level preconditioned Jacobi-Davidson method for eigenvalue problems
Date: 2019.11.22 14:00-14:40
Place: School of Science Room 212
Reporter: JIANG Xuejun
Abstract: In this talk, we shall give a rigorous theoretical analysis of the two-level preconditioned Jacobi-Davidson method for solving the large scale discrete elliptic eigenvalue problems, which was essentially proposed by Zhao, Hwang, and Cai in 2016. Focusing on eliminating the error components in the orthogonal complement space of the target eigenspace, we find that the method could be extended to the case of the 2m th order elliptic operator (m = 1, 2). By choosing a suitable coarse space, we prove that the method holds a good scalability and we obtain the error reduction γ in each iteration, where C is a constant independent of the mesh size h and the diameter of subdomains H, δ = c[1-C(δ2m-1)/(H2m-1)] is the overlapping size among the subdomains, and c → 1 decreasingly as H → 0. Moreover, the method does not need any assumption between H and h. Numerical results supporting our theory are given.
Title 2: Interpolation and Expansion on Orthogonal Polynomials
Date: 2019.11.22 14:40-15:20
Place: School of Science Room 212
Reporter: XIANG Shuhuang
Abstract: The convergence rates on polynomial interpolation in most cases are estimated by Lebesgue constants. These estimates may be overestimated for some special points of sets for functions of limited regularities. In this talk, new formulas on the convergence rates are considered. Moreover, new and optimal asymptotics on the coefficients of functions of limited regularity expanded in forms of Jacobi and Gegenbauer polynomial series are presented. All of these asymptotic analysis are optimal. Numerical examples illustrate the perfect coincidence with the estimates.
Title 3: Spectral methods for some problems having low regularity solutions
Date: 2019.11.22 15:20-16:00
Place: School of Science Room 212
Reporter: XU Chuanju
Abstract: In this talk we will present a new spectral method for a class of equations with non-smooth solutions. The proposed method makes use of the fractional polynomials, also known as Muntz polynomials. We first present some basic approximation properties of the Muntz polynomials, including error estimates for the weighted projection and interpolation operators. Then we will show how to construct efficient spectral methods by using the Muntz polynomials. A detailed convergence analysis will be provided. The potential application of this method covers a large number of problems, including classical elliptic equations, integro-differential equations with weakly singular kernels, fractional differential equations, and so on.
Title 4: Virtual element methods for elliptic variational inequalities of the second kind
Date: 2019.11.22 16:00-16:40
Place: School of Science Room 212
Reporter: HUANG Jianguo
Abstract: In this talk, we are concerned with virtual element methods for solving elliptic variational inequalities (EVIs) of the second kind. First, a general framework is provided for the numerical solution of the EVIs and for its error analysis. Then virtual element methods are applied to solve two representative EVIs: a simplified friction problem and a frictional contact problem. Optimal order error estimates are derived for the virtual element solutions of the two representative EVIs, including the effects of numerical integration for the non-smooth term in the EVIs. A fast solver is introduced to solve the discrete problems. Several numerical examples are included to show the numerical performance of the proposed methods. This is a joint with Fang Feng from Shanghai Jiao Tong University and Weimin Han from University of Iowa.
