Numerische Mathematik

, Volume 136, Issue 1, pp 215–248 | Cite as

Ultraconvergence of high order FEMs for elliptic problems with variable coefficients

  • Wen-ming HeEmail author
  • Zhimin Zhang
  • Qingsong Zou


In this paper, we investigate local ultraconvergence properties of the high-order finite element method (FEM) for second order elliptic problems with variable coefficients. Under suitable regularity and mesh conditions, we show that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed kth \((k\ge 2)\) order finite element solution converges to the gradient of the exact solution with order \(\mathcal{O}(h^{k+2} (\mathrm{ln} h)^3)\). The proof of this ultraconvergence property depends on a new interpolating operator, some new estimates for the discrete Green’s function, a symmetry theory derived in [26], and the Richardson extrapolation technique in [20]. Numerical experiments are performed to demonstrate our theoretical findings.

Mathematics Subject Classification

65N30 65N25 65N15 



The authors would like to thank both the anonymous referees for their careful reading of the paper and their valuable comments which leads to a significant improvement of the paper. The first author is supported in part by the National Natural Science Foundation of China (11671304, 11171257, 11301396), the Zhejiang Provincial Natural Science Foundation, China (No. LY15A010015). The second author is supported in part by the National Natural Science Foundation of China (11471031,91430216) and the US National Science Foundation through grant U1530401. The third author is supported in part by the National Natural Science Foundation of China through grants 11571384 and 11428103, by the Fundamental Research Funds for the Central Universities through grant 16lgjc80, and by Guangdong Provincial Natural Science Foundation of China through grant 2014A030313179.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsWenzhou UniversityWenzhouChina
  2. 2.Beijing Computational Science Research CenterBeijingChina
  3. 3.Department of MathematicsWayne state UniversityDetroitUSA
  4. 4.School of Mathematics and Computational ScienceSun Yat-sen UniversityGuangzhouChina

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