Mesh-Dependent \(L^2\)-Like Norm a Posteriori Error Estimates for Elliptic Problems with Non-essential Boundary Conditions

This work is concerned with the proof of \(L^2\) $L^2$ -like norm residual-type a posteriori error estimates for finite element methods for elliptic problems with non-essential boundary conditions, such as Neumann or Robin type. To ensure the proof of lower bounds (efficiency), a non-standard mesh-dependent \(L^2\) $L^2$ -like norm is used for the error. The proof of lower bounds requires a carefully constructed \(C^1\) $C^1$ -conforming ’bubble’-function. A series of numerical experiments is presented, showcasing the good performance of the estimators.