Finite element methods are not always optimal

Abstract

Consider a regularly elliptic 2mth order boundary value problem Lu = f with f ϵ H r( Ω) , where r $ ̄ m . In previous work, we showed that the finite element method (FEM) using piecewise polynomials of degree k is asymptotically optimal when k ≥ 2 m − 1 + r. In this paper, we show that the FEM is not asymptotically optimal when this inequality is violated. However, there exists an algorithm, called the spline algorithm, which uses the same information as the FEM and is optimal. Moreover, the error of the finite element method can be arbitrarily larger than the error of the spline algorithm. We also obtain a necessary and sufficient condition for a Galerkin method (or a generalized Galerkin method) to be a spline algorithm.