A Decomposition Result for Kirchhoff Plate Bending Problems and a New Discretization Approach

A new approach is introduced for deriving a mixed variational formulation for Kirchhoff plate bending problems with mixed boundary conditions involving clamped, simply supported, and free boundary parts. Based on a regular decomposition of an appropriate nonstandard Sobolev space for the bending moments, the fourth-order problem can be equivalently written as a system of three (consecutively to solve) second-order problems in standard Sobolev spaces. This leads to new discretization methods, which are flexible in the sense that any existing and well-working discretization method and solution strategy for standard second-order problems can be used as a modular building block of the new method. Similar results for the first biharmonic problem have been obtained in our previous work [W. Krendl, K. Rafetseder, and W. Zulehner, Electron. Trans. Numer. Anal., 45 (2016), pp. 257--282]. The extension to more general boundary conditions encounters several difficulties including the construction of an appropriate n...