H²-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 H² and H¹ spaces. To estimate the residual functional acting on H² component, the method reconstructs symmetric and piecewise polynomial tensors on vertex patches in broken H(divdiv) 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, L² 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 H²-based error estimators.