Lower bounds of the discretization error for piecewise polynomials (Q2862516)

This paper is devoted to the study of a lower-bound error estimate and its applications for piecewise polynomial approximation in Sobolev spaces. The authors' work was inspired by some recent studies of lower-bound approximations of eigenvalues by finite element discretization for some elliptic partial differential operators. A type of lower-bound results of the error by piecewise polynomial approximation is proposed. As applications, the authors give the lower bounds of the discretization error for the second-order elliptic and \(2m\)-th-order elliptic problem by finite element methods. Main result: From the analysis, the idea and methods here can be extended to other problems and numerical methods that are based on the piecewise polynomial approximation. The lower bound of the approximation error holds when the family of partitions \(\{\tau_h\}\) is quasi-uniform.NEWLINENEWLINEThe lower bounds for a second-order elliptic problem and the corresponding eigenpair problem by the finite element method are derived. Some conforming and nonconforming elements that yield the lower bound of the discretization error are presented. The authors consider lower bounds of the discretization order for the \(2m\)-th-order elliptic problem and the corresponding eigenpair problem by the finite element method.