Improved $L^2$ and $H^1$ error estimates for the Hessian discretisation method

Devika Shylaja

Published 2018-11-13Version 1

The framework of Hessian discretisation method (HDM) for fourth order linear elliptic equations enables us to develop a study that encompasses numerous classical methods such as finite element methods (conforming and non-conforming), finite volume methods, and others such as methods based on gradient recovery operators. A generic error estimate has been established for the HDM in $L^2$, discrete $H^1$ and $H^2$ norms based on three properties in literature. In this paper, we derive improved $L^2$ and $H^1$ error estimates in the framework of HDM and apply it to various schemes. Since an improved $L^2$ estimate is not expected in general for finite volume method (FVM), here we consider a modified FVM by changing the quadrature of the source term and prove a superconvergence result for this modified FVM. We also show that the non-conforming Morley element fit into the framework of HDM.