研 究 介 绍

本人目前的研究兴趣主要在于偏微分方程数值算法、特征值问题的数值算法、有限元方法、积微分方程的数值算法以及他们在实际问题中的应用，喜欢数值算法的构造及其算法的理论分析。

特征值问题的数值算法

1. 特征值数值算法，我们应该首先从代数特征值问题的求解开始进行理解。因为用有限元方法去离散微分算在特征值所进行的过程其实与代数特征值算法中的子空间投影算法是类似的。我们之后也将会发现，其实有限元特征值的误差估计几乎完全可以按照代数特征值中的子空间投影算法来进行，只是需要在一种合适的形式下进行而已。

我们知道，通常的多重网格算法和区域分解算法是针对线性问题设计的。那么非线性问题呢？

2. 基于对有限元方法中Aubin-Nitsche技巧的一种新的认识，设计了一种求解特征值问题的多重校正算法和多重网格算法。与一般方法不同的是，这种多重网格算法的计算量不依赖于特征值迭代算法的收敛速度，有最优的精度(不仅仅是最优收敛速度)。这种多重网格算法可以应用于非对称特征值问题和非线性特征值问题。

3. 更一般化，多重校正算法可以用来设计求解非线性问题的多重网格算法，比如非线性PDE、控制问题等。

4. 多重校正算法用来设计求解非线性问题的时候可以结合所有的求解线性问题的高效算法，设计出求解非线性问题的高效算法，使得算法的计算量能更进一步摆脱对问题的非线性性质的依赖。

5. 由于能把非线性问题的求解转化成单个线性问题的求解和非常小规模非线性问题的求解，所以更适合用来去设计求解非线性问题的并行算法。

软件下载: FEM_MATLAB (一个可以进行多重校正算法的Matlab软件包).

算法和软件包的介绍可以参看右边这个网格: 介绍网页 

Papers:

2、 Qun Lin and Hehu Xie (谢和虎), A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems, Proceedings of the International Conference Applications of Mathematics 2012, pp. 134-143.

1、Qun Lin and Hehu Xie, An observation on Aubin-Nitsche Lemma and its applications, Mathematics in Practice and Theory, 41(17) (2011), 247-258.(In Chinese, English title and abstract)

Numerical Methods for Partial Differential Equations

1. We prove the lower bound results for the error estimates of the general piecewise polynomial approximations in the general norm sense and on the general meshes. Based on the lower bound results, we derive the necessity of the higher order polynomial interpolation in the superconvergence and extrapolation methods.

2. The lower bound results also provide a mathematical tool to prove the asymptotic lower bound results of the eigenvalues by some nonconforming finite element methods.

Papers:

7. Shanghui Jia, Fusheng Luo and Hehu Xie, A Posterior Error Analysis for the Nonconforming Discretization of Stokes Eigenvalue Problem, Acta Mathematica Sinica, English Series, 30(6) (2014), 949–967.

6. Qun Lin, Fusheng Luo and Hehu Xie, A posteriori error estimator and lower bound of a nonconforming finite element method, Journal of Computational and Applied Mathematics, 256(1), pages 243-254, 2014.

5. Qun Lin and Hehu Xie, Recent results on Lower bounds of eigenvalue problems by nonconforming finite element methods, Inverse Problems and Imaging, Volume 7, No. 3, 2013, 795-811.

4.   Qin Li, Qun Lin and Hehu Xie,Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations, Applications of Mathematics, Volume 58, Issue 2, 2013, pp 129-151.

3 F.Luo, Qun Lin and Hehu Xie (谢和虎), Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods, Science China Mathematics, Volume 55, Issue 5, pp 1069-1082, 2012.

2、Q. Lin and H. Xie, The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods, Mathematics in Practice and Theory, 42 (11) (2012), 219–226.(In Chinese with English title and abstract)

1、Q. Lin, H. Xie, F. Luo, Y. Li and Y. Yang, Stokes eigenvalue approximations from below with nonconforming mixed finite element methods, Mathematics in Practice and Theory, 40(19)(2010), 157-168. (In Chinese, English title and abstract)

Numerical Method for Phase Field Models

1. Convergence theory for spectral deferred correction method: We give the convergence analysis for the SDC method.

2. Numerical method for phase field methods (time adaptivity, mixed finite element method).

3. Numerical method for convection-diffusion-reaction equations.

4. Method to define the metric tensor.

Papers:

34. Yu Li, Qun Lin and Hehu Xie, A parallel method for population balance equations based on the method of characteristics, Proceedings of the International Conference Applications of Mathematics, 2013, pp. 140-149.

32. Tao Tang, Hehu Xie and Xiaobo Yin, High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes, Journal of Scientific Computing  Volume 56, Issue 1, 2013, pp 1-13

36. Xiaobo Yin and Hehu Xie, Metric tensors for the interpolation error and its gradient in $L^p$ norm, Journal of Computational Physics, 256(1), Pages: 543-562, 2014.

52.    Zhonghua Qiao, Tao Tang and Hehu Xie, Error analysis of a mixed finite element method for molecular beam epitaxy model, SIAM J. Numer. Anal., 53(1) (2015), 184-205.

56. Fusheng Luo, Tao Tang and Hehu Xie, Parametert-free time adaptivity based on energy evolution for Cahn-Hilliard equation, Commun. Comput. Phys., 19(05) (2016), 1542-1563.

16. N. Ahmed, G. Matthies, L. Tobiska and H. Xie, Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reaction problems，Comput. Methods Appl. Mech. Engrg., 200 (2011), 1747-1756.

Two-grid method for eigenvalue problems

Papers:

3.   Hehu Xie and Xiaobo Yin, Acceleration of stabilized finite element discretizations for the Stokes eigenvalue problem, Adv. Comput. Math., 41(2015), 799–812.

2. X. Feng, Z, Wen and Hehu Xie, Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem, Application of Mathematics, 59(6) (2014), 615-630.

1.     Hongtao Chen, Shanghui Jia and Hehu Xie, Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems, Application of Mathematics, 54(3)(2009), 237-250.

Numerical Method for Differential-Integral Equations and Ordinary Differential Equations　

1. Functional equation and delay differential equation: We study the collocation method for a class of functional equations with vanish delays and the corresponding Volterra functional integral equations, the discontinuous Galerkin method for the delay differential equations of pantograph type and the corresponding interesting superconvergence, the hp discontinuous Galerkin (hp-DG) method for delay differential equations with nonlinear delay.

Papers:

7.   Ran Zhang, Benxi Zhu and Hehu Xie, Spectral methods for weakly singular Volterra integral equations with   pantograph delays，Frontiers of Mathematics in China,  Vol. 8, Issue 2, 2013, pp 281-299

6.   Qiumei Huang, Hehu Xie and Hermann Brunner, The $hp$ Discontinuous Galerkin Method for Delay Differential Equations with Nonlinear Vanishing Delay, SIAM J. Sci. Comput., 35(3),  2013, A1604–A1620.

5. H. Xie, R. Zhang and H. Brunner, Collocation Methods for General Volterra Functional Integral Equations with Vanishing Delays, SIAM Journal on Scientific Computing, 33(6) (2011), 3303-3332.

4. Q. Huang, H. Xie and H. Brunner, Superconvergence of a discontinuous Galerkin solutions for delay differential equations of pantograph type, SIAM Journal on Scientific Computing， 33(5) (2011),  2664–2684.

3、   H. Brunner, Hehu Xie and R. Zhang, Analysis of collocation solutions for a class of functional equations with vanishing delays,    IMA Journal of Numerical Analysis, 31(2011), 698-718.

2、H. Brunner, Q. Huang and H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. NUMER. ANAL., Vol. 48, No. 5, pp. 1944–1967, 2010.

1.   Q. Huang and Hehu Xie, Superconvergence of Galerkin solutions for Hammerstein equations, International Journal of Numerical Analysis & Modeling, 2009， Volume 6, Number 4, Pages: 696-710.

Finite Element Methods: Accuracy and Improvement 

1. We give asymptotic eigenvalue error expansions and extrapolation schemes for the Laplace eigenvalue problem on the classical four types of uniform triangular meshes.

2. We analyze the asymptotic error expansions and extrapolation for the second order elliptic eigenvalue problem and Stokes eigenvalue problem by the mixed finite element method.

3. We analyze the superconvergence and extrapolation on the general triangular meshes.

4. We analyze the superconvergence of the Stokes problem by local projection stabilization methods.

Papers:

16.  Shanghui Jia， Hongtao Chen and Hehu Xie, A posteriori error estimator for eigenvalue problems by mixed finite element method, Science China Mathematics,Volume 56, Issue 5, 2013, pp 887-900

15.  Qun Lin and Hehu Xie, Extrapolation of the finite element method on general meshes, International Journal of Numerical Analysis and Modeling Computing and Information, Vol. 10, Number 1, 2013, pp. 139-153.

14. Qun Lin and Hehu Xie (谢和虎), A Superconvergence result for mixed finite element approximations of the eigenvalue problem^, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 46/Issue 04,  pp 797-812, 2012.

13. Qun Lin and Hehu Xie, A type of finite element gradient recovery method based on vertex-edge-face interpolation: The recovery rechnique and superconvergence property, East Asian Journal on Applied Mathematics, 1(3) (2011), 248-263.

12. S. Jia, H. Xie and X. Yin, A type of finite element gradient recovery method based on vertex-edge interpolation, Mathematics in Practice and Theory, 41(6)  (2011),  239-243. (In Chinese, English title and abstract)

11.   Qun Lin and Hehu Xie, Superconvergence measurement for General meshes by linear finite element method, Mathematics in Practice and Theory, 41(1)(2011), 138-152. (In Chinese, English title and abstract)

10. H. Eichel, L. Tobiska and H. Xie, Supercloseness and superconvergence of stabilized low order finite element discretizations of the Stokes problem, Mathematics of Computation, 80(274)(2011), 697–722.

9.   Qun Lin and Hehu Xie, New expansions of numerical eigenvalue for $-\Delta u = \lambda\rho u$ by linear finite element on different triangular meshes, International Journal of Information and Systems Sciences, 6(1)(2010), 10-34

8.  Qun Lin and Hehu Xie, Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method, Applied Numerical Mathematics, 59(8)(2009, 1884-1893.

7.   S. Jia, Hehu Xie, X. Yin and S. Gao, Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods, Applications of Mathematics, 54(1)(2009), 1-15.

6.     Hehu Xie and S. Jia, Extrapolation for the second order elliptic problems by mixed finite element methods in three dimensions,  International Journal of Numerical Analysis and Modeling, 5(1)(2008), 112-131.

5.       S. Jia, Hehu Xie, X. Yin and S. Gao, Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods,  Numerical Methods for Partial Differential Equations, 24(2) ,435-448,2008.

4.       X. Yin, Hehu Xie, S. Jia and S. Gao, Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods, Journal of Computational and Applied Mathematics, 215(1)(2008), 127-141

3.        Hehu Xie, Extrapolation of the Nédélec element for the Maxwell equations by the mixed finite element method, Advances in Computational Mathematics, 29(2) (2008), 135-145. 

2.        Hehu Xie and S. Gao, Superconvergence of the least-squares mixed finite element approximations for the second order elliptic problems, International Journal of Information and Systems Sceiences, Vol.3 (2) (2007), 277-282.

1.       V. Shaidurov and Hehu Xie, Richardson extrapolation for eigenvalue of discrete spectral problem on general mesh, Computational Technologies, Vol. 12 No. 3(2007), 24-37.