$H^2$-based equilibrated residual error estimates

We develop a framework for equilibrated residual a posteriori error analysis of finite element methods, based on decomposing vector- and tensor-valued Sobolev spaces into $H^2$ and $H^1$ spaces. To estimate the residual functional acting on $H^2$ component, the method reconstructs symmetric and piecewise polynomial tensors on vertex patches in broken $H(\operatorname{div}\operatorname{div})$ spaces. We apply the proposed framework to conforming finite element discretizations of biharmonic equation, Poisson's equation, and linear elasticity in mixed form. As a result, we obtain several novel equilibrated residual a posteriori error estimates, providing guaranteed error control for energy error of biharmonic equation, $L^2$ norm error of Poisson's equation, and stress error of mixed linear elasticity. They also serve as the first polynomial-degree-robust a posteriori lower bounds of discretization errors. Numerical experiments illustrate the effectiveness of the $H^2$-based error estimators.