Qun Lin

Qun Lin, Hehu Xie, Jinchao Xu

Assume that $V_h$ is a space of piecewise polynomials of degree less than $r\geq 1$ on a family of quasi-uniform triangulation of size $h$. Then the following well-known upper bound holds for a sufficiently smooth function $u$ and $p\in [1, \infty]$ $$ \inf_{v_h\in V_h}\|u-v_h\|_{j,p,Ω,h} \le C h^{r-j} |u|_{r,p,Ω},\quad 0\le j\le r. $$ In this paper, we prove that, roughly speaking, if $u\not\in V_h$, the above estimate is sharp. Namely, $$ \inf_{v_h\in V_h}\|u-v_h\|_{j,p,Ω,h} \ge c h^{r-j},\quad 0\le j\le r, \ \ 1\leq p\leq \infty, $$ for some $c>0$. The above result is further extended to various situations including more general Sobolev space norms, general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.

Qun Lin, Hehu Xie

In this paper, a new type of multi-level correction scheme is proposed for solving eigenvalue problems by finite element method. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving eigenvalue problems by finite element method.

Fusheng Luo, Qun Lin, Hehu Xie

This article is devoted to computing the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extension $Q_1^{\rm rot}$, we get the lower bound of the eigenvalue. Additionally, we also use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue. The postprocessing method need only to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to validate our theoretical analysis.

Jun Hu, Yunqing Huang, Qun Lin

The aim of the paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The general conclusion herein is that if local approximation properties of nonconforming finite element spaces $V_h$ are better than global continuity properties of $V_h$, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we first show abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. As one application, we show that this condition hold for most nonconforming elements in literature. As another important application, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.

Wujian Peng, Qun Lin

Most current prevalent iterative methods can be classified into the so-called extended Krylov subspace methods, a class of iterative methods which do not fall into this category are also proposed in this paper. Comparing with traditional Krylov subspace methods which always depend on the matrix-vector multiplication with a fixed matrix, the newly introduced methods(the so-called (progressively) accumulated projection methods, or AP (PAP) for short) use a projection matrix which varies in every iteration to form a subspace from which an approximate solution is sought. More importantly an accelerative approach(called APAP) is introduced to improve the convergence of PAP method. Numerical experiments demonstrate some surprisingly improved convergence behavior. Comparison between benchmark extended Krylov subspace methods(Block Jacobi and GMRES) are made and one can also see remarkable advantage of APAP in some examples. APAP is also used to solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix) and numerical experiments shows that it can bring very satisfactory results even when the size of system is up to a few thousands.

Yu Li, Qun Lin, Hehu Xie

In this paper, we present a parallel scheme to solve the population balance equations based on the method of characteristics and the finite element discretization. The application of the method of characteristics transform the higher dimensional population balance equation into a series of lower dimensional convection-diffusion-reaction equations which can be solved in a parallel way.Some numerical results are presented to show the accuracy and efficiency.