Lowest-order equivalent nonstandard finite element methods for biharmonic plates

Abstract

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side F ∈ H−2(Ω) replaced by F ○ (JIM) and then are quasi-optimal in their respective discrete norms. The smoother JIM is defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj, ESAIM: M2AN 49 (2015) 977–990.] without data oscillations. This paper extends the work [Veeser and Zanotti, SIAM J. Numer. Anal. 56 (2018) 1621–1642] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.