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Research topics

• Guaranteed eigenvalue estimation for differential operators

The finite element method is employed to derive precise bounds for the eigenvalues of differential operators. Historically, establishing upper bounds for eigenvalues has been successfully achieved. However, providing lower bounds for eigenvalues remains a significant challenge. In contrast to traditional eigenvalue analysis, which predominantly concentrates on efficient algorithms for eigenvalue approximations, recent advancements have fostered the development of novel theories and computational techniques to ascertain both rigorous lower and upper bounds for these eigenvalues.

The Lehmann-Goerisch theorem plays a pivotal role in obtaining high-precision bounds for eigenvalues using conforming approximate eigenfunctions, such as those derived from conforming finite element methods (FEMs).

• Quantitative error analysis for the finite element method

The traditional error analysis pertaining to the finite element method primarily focuses on qualitative aspects, such as the convergence rates of numerical schemes. However, in numerous scenarios, such as when seeking a computer-assisted mathematical proof, it is imperative to obtain precise values or upper bounds of approximation errors. Addressing this necessitates confronting the subsequent challenges, which are intrinsically linked to the eigenvalue problem of differential operators:

• Providing not only boundedness but also explicit values for various error constants within the error analysis.
• Addressing the singularity present in partial differential equation (PDE) solutions.

• Computer-assisted mathematical proofs via verified computation

Verified computation seeks to address numerous challenges in order to deliver rigorous computational results, which can even be incorporated into mathematical proofs.

To achieve this, interval arithmetic is employed to handle the rounding errors arising from computations with floating-point numbers. Additionally, the development of novel computational algorithms and error analysis theories is essential to assess various errors, for example, the truncation error and the function approximation error.

A significant area within verified computation involves exploring the existence and uniqueness of solutions to non-linear partial differential equations, such as the Navier-Stokes equation.

• Resistivity measurement via the four-probe method

  
  Supported by JKA RING!RING! Project.
  競輪・オートレース補助事業

  The four-probe method serves as an efficient approach to gauging the resistivity of semiconductor materials. The underlying mathematical model is predicated upon the subsequent Poisson boundary value problem:

  \( -\Delta u = f \text{ in } \Omega, {\partial{u}}/{{\partial n}} =0 \text{ on } \partial \Omega . \)

  By resolving the aforementioned problem within a 3D domain and addressing the singularity in the solution, one can derive the resistivity. For a more comprehensive discussion, please refer to the provided details (URL).

  • This is research is supported by JKA RING!RING! Project.

Papers

Journal papers

• See a full list of papers at Google Scholar Citations Report .
  
• [Before 2018]

1. Qin LI, Xuefeng LIU, Explicit Finite Element Error Estimates for Nonhomogeneous Neumann problems, Applications of Mathematics, Vol. 63, Issue 3, pp. 367-379, 2018. Link at arxiv.org, 2018/06/05.
2. Xuefeng LIU, Fumio KIKUCHI, Explicit Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element, Applications of Mathematics, Vol. 63, Issue 4, pp. 381–39, 2018. Link at arxiv.org, 2018/05/27.
3. Manting Xie, Hehu Xie, Xuefeng Liu, Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements, Vol. 35, Issue 1, pp. 335–354, 2018, March (link pdf)
4. Xuefeng Liu, Chun'guang You, Explicit Bound for Quadratic Lagrange Interpolation Constant on Triangular Finite Elements, Applied Mathematics and Computation, Vol.319, pp. 693-701, 2018. (arXiv URL)
5. Xuefeng LIU, Yuan LIU, and Zhouwang Yang, Paper review assignment problem under the minimum cost flow network model (ネットワークフローのモデルで論文査読者割当の問題を考える), Proceedings of the Twenty-Eighth RAMP Symposium, Niigata University, Niigata, October 13–14, 2016. PDF
6. Xuefeng Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Applied Mathematics and Computation, 267, pp.341–355, 2015 (PDF)
7. Xuefeng Liu, Tomoaki Okayama and Shin’ichi Oishi High-Precision Eigenvalue Bound for the Laplacian with Singularities, Editors: Ruyong Feng, Wen-shin Lee and Yosuke Sato, Computer Mathematics, 311-323, Spinger, 2014. (PDF)
8. Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu and Shin'ichi Oishi, Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation, Japan J. Indust. Appl. Math. 31, pp.665–679, 2014 (PDF)
9. Akitoshi Takayasu, Xuefeng Liu and Shin'ichi Oishi,Remark on computable a priori error estimation for higher degree finite element solution of elliptic problem, NOLTA, IEICE, Vol.5, No.1, pp.53-63, 2014. PDF
10. Xuefeng Liu and Shin'ichi Oishi, Guaranteed high-precision estimation for P0 interpolation constants on triangular finite elements, Japan Journal of Industrial and Applied Mathematics 30 (3), 635-652 PDF
11. Xuefeng Liu and Shin'ichi Oishi, Verified eigenvalue evaluation for Laplacian over polygonal domain of arbitrary shape, SIAM J. Numer. Anal., 51(3), 1634-1654. (link PDF)
12. Xuefeng Liu and Shin'ichi Oishi, Verified eigenvalue evaluation for Laplace operator on arbitrary polygonal domain, RIMS Kokyuroku, 2011. (link, PDF)
13. A. Takayasu, X. Liu and S. Oishi, Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains, NOLTA, IEICE, Vol.E96-N, No.1, pp.34-61, Jan. 2013. (PDF)
14. Xuefeng Liu and Fumio Kikuchi, Analysis and estimation of error constants for P0 and P1 interpolations over triangular finite elements, J. Math. Sci. Univ. Tokyo, 17, p.27-78, 2010 (PDF)
15. Fumio Kikuchi and Xuefeng Liu, Estimation of interpolation error constants for the P0 and P1 triangular finite elements, Computer Methods in Applied Mechanics and Engineering, Vol.196, Issues 37-40, (2007), pp. 3750-3758. PDF
16. Xuefeng Liu and Fumio Kikuchi, Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element, Proceeding of APCOM’07 in conjunction with EPMESC XI (digital version only), December 3-6, 2007, Kyoto, JAPAN. PDF
17. Fumio Kikuchi and Xuefeng Liu, Determination of the Babuska-Aziz constant for the linear triangular finite element, Japan Journal of Industrial and Applied Mathematics (JJIAM), Vol.23, No.1, (2006), pp. 75-82. PDF

(Paper finished in USTC, China.)
18. Ghulam Mustafa and Xuefeng Liu, A subdivision scheme for volumetric models. Appl. Math.　J. Chinese Univ. Ser. B, Vol 20, no.2, p.213–224, 2005
19. Ghulam Mustafa and Xuefeng Liu, A New Solid Subdivision Scheme, Journal of University of Science and Technology of China, Vol.35(3), p.285-300, 2005
20. Xuefeng Liu, Blending Pipe Surface by Ringed Surface and Its Continuity, The Journal of　Uni. of Sci. & Tech. of China, Vol.34 No.1, P.20-P28, 2004
21. Xuefeng Liu, Jin Xu and Ronggang Bai, 3D Rebuilding of Vessel, Chinese Journal of Engineering Mathematics, Vol. 19 Supp., p.35-40, 2002.

学位論文

1. 博士論文： 適合および非適合1 次有限要素の誤差定数の解析(Analysis of error constants for linear conforming and nonconforming finite elements)　東京大学数理科学研究科2009年3月 （指導教官：菊地文雄） ( PDF download )
2. 修士論文：０次および１次三角形有限要素の補間誤差定数の評価 (Estimation of error constants for $P_0$ and $P_1$ interpolations over triangular finite elements)　東京大学数理科学研究科2006年3月（指導教官：菊地文雄）