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 0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side ∈ −2 (Ω) replaced by • ( M ) and then are quasi-optimal in their respective discrete norms. The smoother M is defined for a piecewise smooth input function by a (generalized) Morley interpolation M followed by a companion operator . 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: Comparison results of nonstandard 2 finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends the work [Veeser, Zanotti: Quasioptimal nonconforming methods for symmetric elliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

$C^1$-conforming variational discretization of the biharmonic wave equation

“…We note that the finite element approximation of the biharmonic operator continues to be an active field of research, for recent contributions, see, e.g. [21,22,25]. In particular, discretization methods that support polyhedral meshes (the mesh cells can be polyhedra or have a simple shape but contain hanging nodes) and hinge on the primal formulation of the biharmonic equation leading to a symmetric positive definite system matrix are currently focused.…”