Lowest-order equivalent nonstandard finite element methods for biharmonic plates

TL;DR

This paper develops and analyzes lowest-order nonstandard finite element methods for biharmonic plates, extending existing frameworks to new schemes and providing quasi-optimal error estimates in various norms.

Contribution

It introduces a unified abstract error analysis framework for nonstandard finite element methods for biharmonic equations, including new error estimates for the discontinuous Galerkin scheme.

Findings

Methods are quasi-optimal in their discrete norms.

Error estimates are extended to weaker and piecewise Sobolev norms.

Three errors are shown to be equivalent in specific discrete norms.

Abstract

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the $C^0$ interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side $F\in H^{-2}(\Omega)$ replaced by $F\circ (JI_{\rm M}) $ and then are quasi-optimal in their respective discrete norms. The smoother $JI_{\rm M}$ is defined for a piecewise smooth input function by a (generalized) Morley interpolation $I_{\rm M}$ 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: Comparison results of nonstandard $P_2$ finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends the work [Veeser, Zanotti: Quasi-optimal 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.