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Hehu Xie

Publications and source records attributed to Hehu Xie.

77 records · Page 5Linked to original sources
arXiv2011

A Multi-level Correction Scheme for Eigenvalue Problems

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.

math.NA↗
arXiv2011

Lower Bounds of the Discretization for Piecewise Polynomials

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.

math.NA↗
arXiv2011

New metric tensors for anisotropic mesh generation

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on a posteriori error estimates proposed in \cite{YinXie}. The new metric tensor explores more comprehensive information of anisotropy for the true solution than those existing ones. Numerical results show that this approach can be successfully applied to deal with poisson and steady convection-dominated problems. The superior accuracy and efficiency of the new metric tensor to others is illustrated on various numerical examples of complex two-dimensional simulations.

math.NA↗
arXiv2006

Asymptotic Expansions and Extrapolation of Approximate Eigenvalues for Second Order Elliptic Problems by Mixed Finite Element Methods

In this paper, we derive an asymptotic error expansion for the eigenvalue approximations by the lowest order Raviart-Thomas mixed finite element method for the general second order elliptic eigenvalue problems. Extrapolation based on such an expansion is applied to improve the accuracy of the eigenvalue approximations. Furthermore, we also prove the superclose property between the finite element projection with the finite element approximation of the eigenvalue problems by mixed finite element methods. In order to prove the full order of the eigenvalue extrapolation, we first propose "the auxiliary equation method". The result of this paper provides a general procedure to produce an asymptotic expansions for eigenvalue approximations by mixed finite elements.

math.NA↗