The hellan-herrmann-johnson method with curved elements

Douglas N. Arnold; Shawn W. Walker

doi:10.1137/19M1288723

The hellan-herrmann-johnson method with curved elements

Douglas N. Arnold, Shawn W. Walker

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 C^k⁺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 language	English (US)
Pages (from-to)	2829-2855
Number of pages	27
Journal	SIAM Journal on Numerical Analysis
Volume	58
Issue number	5
DOIs	https://doi.org/10.1137/19M1288723
State	Published - Oct 14 2020

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. https://doi.org/10.1137/19M1288723 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@ umn.edu, http://umn.edu/∼arnold/). ‡Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918 USA (walker@math.lsu.edu, http://www.math.lsu.edu/∼walker/).

Keywords

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

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title = "The hellan-herrmann-johnson method with curved elements",

abstract = "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 Babu{\v s}ka paradox, highlighting the geometrically nonconforming aspect of the HHJ method.",

keywords = "Babu{\v s}ka paradox, Geometric consistency error, Kirchhoff plate, Mesh-dependent norms, Parametric finite elements, Simply supported",

author = "Arnold, {Douglas N.} and Walker, {Shawn W.}",

note = "Funding Information: ∗Received by the editors September 20, 2019; accepted for publication (in revised form) July 21, 2020; published electronically October 13, 2020. https://doi.org/10.1137/19M1288723 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@ umn.edu, http://umn.edu/∼arnold/). ‡Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918 USA (walker@math.lsu.edu, http://www.math.lsu.edu/∼walker/).",

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N2 - 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.

AB - 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.

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