Geodesic
A decomposition result for biharmonic problems and the Hellan-Herrmann-Johnson method
Electronic transactions on numerical analysis, Tome 45 (2016), pp. 257-282
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.
Classification : 65N22, 65F10, 65N55
Keywords: biharmonic equation, hellan-herrmann-Johnson method, mixed methods, Helmholtz decomposition 
@article{ETNA_2016__45__a12,
     author = {Krendl,  Wolfgang and Rafetseder,  Katharina and Zulehner,  Walter},
     title = {A decomposition result for biharmonic problems and the {Hellan-Herrmann-Johnson} method},
     journal = {Electronic transactions on numerical analysis},
     pages = {257--282},
     year = {2016},
     volume = {45},
     zbl = {1347.65169},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2016__45__a12/}
}
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%A Zulehner,  Walter
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%J Electronic transactions on numerical analysis
%D 2016
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%F ETNA_2016__45__a12
Krendl,  Wolfgang; Rafetseder,  Katharina; Zulehner,  Walter. A decomposition result for biharmonic problems and the Hellan-Herrmann-Johnson method. Electronic transactions on numerical analysis, Tome 45 (2016), pp. 257-282. http://geodesic.mathdoc.fr/item/ETNA_2016__45__a12/