A Decomposition Result for Biharmonic Problems and the Hellan-herrmann-johnson Method a Decomposition Result for Biharmonic Problems and the Hellan-herrmann-johnson Method *

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 algorithm for the Hellan-HerrmannJohnson method is that it is solely based on standard Lagrangian finite element spaces and standard multigrid methods for second-order elliptic problems and it is of optimal complexity.

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