Quasi-interpolation and best approximation

Abstract

One of the objectives of this chapter is to estimate the decay rate of the best-approximation errors of functions in Sobolev spaces by members of conforming finite element spaces. The interpolation operators constructed so far do not give a satisfactory answer to the above question when the functions have a low smoothness index. In this chapter, we introduce the important notion of quasi-interpolation, i.e., we build linear operators that are \(L^1\)-stable, are projections onto conforming finite element spaces, and have optimal local approximation properties. We do this by composing one of the \(L^1\)-stable operators onto the larger broken finite element space with a simple averaging operator. We also adapt the construction to enforce zero traces at the boundary. We finally study the approximation properties of the \(L^2\)-orthogonal projection onto the conforming finite element spaces. The material of this chapter is important to investigate the approximation of solutions to PDEs with low regularity.