Enforcing Neumann Boundary Conditions with Polynomial Extension Operators to Acheive Optimal Convergence Rates on Polytopial Meshes in the Finite Element Method

In \cite{cheung2019optimally}, the authors presented two finite element methods for approximating second order boundary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear mappings. This was done by enforcing the boundary conditions through judiciously chosen polynomial extension operators. The $H^1$ error estimates were proven to be optimal for the solutions of both the Dirichlet and Neumann boundary value problems. It was also proven that the Dirichlet problem approximation converges optimally in $L^2$. However, optimality of the Neumann approximation in the $L^2$ norm was left as an open problem. In this work, we seek to close this problem by presenting new analysis that proves optimal error estimates for the Neumann approximation in the $W^1_\infty$ and $L^2$ norms.