Virtual Element Method for fourth order problems: $L^2-$estimates

TL;DR

This paper analyzes a family of $C^1$-Virtual Elements for fourth-order problems, providing optimal $L^2$ and $H^1$ estimates using duality arguments, advancing numerical methods for complex PDEs.

Contribution

It introduces and proves optimal $L^2$ and $H^1$ estimates for $C^1$-Virtual Elements applied to fourth-order problems, extending previous work with rigorous analysis.

Findings

Optimal $L^2$ estimates established

Optimal $H^1$ estimates proven

Duality arguments effectively used for analysis

Abstract

We analyse the family of $C^1$-Virtual Elements introduced in \cite{Brezzi:Marini:plates} for fourth-order problems and prove optimal estimates in $L^2$ and in $H^1$ via classical duality arguments.