Volume 45, pp. 257-282, 2016.

A decomposition result for biharmonic problems and the Hellan-Herrmann-Johnson method

Wolfgang Krendl, Katharina Rafetseder, and Walter Zulehner


For the first biharmonic problem a mixed variational formulation is introduced which is equivalent to a standard primal variational formulation on arbitrary polygonal domains. Based on a Helmholtz decomposition for an involved nonstandard Sobolev space it is shown that the biharmonic problem is equivalent to three (consecutively to solve) second-order elliptic problems. Two of them are Poisson problems, the remaining one is a planar linear elasticity problem with Poisson ratio 0. The Hellan-Herrmann-Johnson mixed method and a modified version are discussed within this framework. The unique feature of the proposed solution algorithms for the Hellan-Herrmann-Johnson method and its modified variant is that they are solely based on standard Lagrangian finite element spaces and standard multigrid methods for second-order elliptic problems and that they are of optimal complexity.

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Key words

biharmonic equation, Hellan-Herrmann-Johnson method, mixed methods, Helmholtz decomposition

AMS subject classifications

65N22, 65F10, 65N55

Links to the cited ETNA articles

[15]Vol. 37 (2010), pp. 214-238 Susanne C. Brenner, Thirupathi Gudi, and Li-Yeng Sung: A weakly over-penalized symmetric interior penalty method for the biharmonic problem
[25]Vol. 22 (2006), pp. 1-16 M. R. Hanisch: Two-level additive Schwarz preconditioners for fourth-order mixed methods
[39]Vol. 17 (2004), pp. 112-132 Jie Zhao: Convergence of V-cycle and F-cycle multigrid methods for the biharmonic problem using the Morley element

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