Research Introduction

 

My research areas are numerical methods for partial differential equations, numerical methods for eigenvalue problem, finite element method, numerical methods for differential-integral problems and theirs applications. I am especially interested in the numerical method designing and theirs theoretical analysis.


Numerical Methods for Eigenvalue Problems

1. It is well known that the classical multigrid method and domain decomposition method is designed for linear problems. How about nonlinear problems?

2. Based on a new interesting understanding of the Aubin-Nitsche technique in the finite element theory, we develop a type of multilevel correction method for the eigenvalue problem. This multilevel correction method can transform the eigenvalue problem solving to a boundary value problem solving and small scale eigenvalue problems solving in a very coarse finite dimensional space.

3. The multilevel correction method provides a reasonable way to design a multigrid method for nonlinear problems, linear and nonlinear eigenvalue problems.

4. The multilevel correction method can also be used to design the multigrid method for other nonlinear problems such as semilinear problem and optimal controls of elliptic equations.

5. The multilevel correction method can couple with other efficient numerical methods for boundary value problems to design the efficient solves for eigenvalue problems, semilinear problems and so on.

6. The multilevel correction method can provide an idea to design the parallel method for nonlinear problems.

SoftWare: FEM_MATLAB (A Matlab software can do the multilevel correction method)

If anyone have problem, please send me by email.

An introduction page for this method and software: Introduction Page

 

Papers:

18. Xiaole Han, Hehu Xie and Fei Xu, A cascadic multigrid method for eigenvalue problem, Journal of Computational Mathematics, 35(1) (2017), 7-90. 

17. Shanghui Jia, Hehu Xie, Manting Xie and Fei Xu, A full multigrid method for nonlinear eigenvalue problems, Sci China Math, 59 (2016), 2037–2048.

16. Hehu Xie and Manting Xie, A multigrid method for ground state solution of Bose-Einetein condensates, Commun. Comput. Phys., 19(3) (2016), 648-662.

15. Hongtao Chen, Hehu Xie and Fei Xu, A full multigrid method for eigenvalue problems, Journal of Computational Physics,322 (2016),747-759.

14. Xiaole Han, Yu Li, Hehu Xie and Chunguang You, Local and parallel finite element algorithm based on multilevel discretization for eigenvalue problems, International Journal of Numerical Analysis & Modeling, 13(1) (2016),73-89.

13.    Hehu Xie and Tao Zhou, A multilevel finite element method for Fredholm integral eigenvalue problems,Journal of Computational Physics,303 (2015),173-184

12.    Hongtao Chen,Yunhui He, Yu Li and Hehu Xie, A multigrid method for eigenvalue problems based on shifted-inverse power technique. European Journal of Mathematics, 1(1) (2015), 207-228.

11.    Xiaole Han, Yu Li and Hehu Xie, A multilevel correction method for Steklov eigenvalue problem by nonconforming finite element methods, Numerical Mathematics: Theory, Methods and Applications, 8(03) (2015), 383-405.

10.    谢和虎, 非线性特征值问题的多重网格算法,中国科学:数学,458) (2015),1193-1204.

      Hehu Xie, A multigrid method for nonlinear eigenvalue problems (in Chinese), Sci Sin Math, 45 (2015), 1193-1204

9.    Qun Lin, Hehu Xie and Fei Xu, Multilevel correction adaptive finite element method for semilinear elliptic equation, Application of Mathematics, 60(5) (2015), 527-550.

8. Wei Gong, Hehu Xie, Ningning Yan, A multilevel correction method for optimal controls of elliptic equation, SIAM J. Sci. Comput., 37(5) 2015, A2198-A2221, 2015.

7. Qun Lin, Fusheng Luo and Hehu Xie, A multilevel correction method for Stokes eigenvalue problems and its application, Mathematical Methods in the Applied Sciences, 38(18) (2015), 4540-4552.

6. Qun Lin and Hehu Xie, A multi-level correction scheme for eigenvalue problems, Mathematics of Computation, 84(291) (2015), 71-88.

5. Hehu Xie, A type of multilevel method for the Steklov eigenvalue problem, IMA J. Numer. Anal., 34(2) (2014), 592-608.

4. Hehu Xie, A multigrid method for eigenvalue problem, Journal of Computational Physics, 274(1), 2014, pages 550-561.

3. Xia Ji, Jiguang Sun and Hehu Xie, A multigrid method for Helmholtz transmission eigenvalue problems, Journal of Scientific Computing, 60(2), pages: 276-294, 2014. 

2Qun 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.

1Qun 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:

8. Qun Lin, Hehu Xie and Jinchao Xu, Lower Bounds of the Discretization for Piecewise Polynomials, Math. Comp., 83 (2014), 1-13.

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)

1Q. 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, Parameter-free time adaptivity based on energy evolution for the 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-reactionproblemsComput. 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 delaysFrontiers 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.

2H. 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.

13Qun 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.

12S. 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), 697722.

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.