The hellan-herrmann-johnson method with curved elements

Douglas N. Arnold, Shawn W. Walker

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise Ck+1 boundary for k ≥ 1 when using a parametric (curved) HHJ space. Computational results are given that demonstrate optimal convergence and how convergence degrades when curved triangles of insufficient polynomial degree are used. Moreover, we show that the lowest order HHJ method on a polygonal approximation of the disk does not succumb to the classic Babuška paradox, highlighting the geometrically nonconforming aspect of the HHJ method.

Original languageEnglish (US)
Pages (from-to)2829-2855
Number of pages27
JournalSIAM Journal on Numerical Analysis
Issue number5
StatePublished - Oct 14 2020
Externally publishedYes

Bibliographical note

Funding Information:
∗Received by the editors September 20, 2019; accepted for publication (in revised form) July 21, 2020; published electronically October 13, 2020. Funding: The work of the first author was supported by National Science Foundation grant DMS-1719694 and by Simons Foundation grant 601397. The work of the second author was supported by National Science Foundation grant DMS-155222. †Department of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (arnold@,∼arnold/). ‡Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918 USA (,∼walker/).

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics


  • Babuška paradox
  • Geometric consistency error
  • Kirchhoff plate
  • Mesh-dependent norms
  • Parametric finite elements
  • Simply supported


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