Commuting quasi-interpolation

Abstract

The quasi-interpolation operators introduced in Chapter 22 are \(L^1\)-stable, are projections and have optimal (local) approximation properties. However, they do not commute with the usual differential operators (gradient, curl, and divergence), which makes them difficult to use to approximate simultaneously a vector-valued function and its curl or its divergence. Since these commuting properties are important in some applications, we introduce in this chapter quasi-interpolation operators that are \(L^1\)-stable, are projections, have optimal (global) approximation properties, and have the expected commuting properties. The key idea is to compose the canonical interpolation operators with mollification operators, i.e., smoothing operators based on the convolution with a smooth kernel.