Interpolation operators with stability properties are key for error control in numerical analysis. Recent applications including multilevel decomposition and a posteriori error control for rough right-hand sides require stability in dual Sobolev spaces, that most standard operators do not satisfy.
We present a Scott–Zhang type projection operator mapping to continuous Lagrange elements for arbitrary polynomial degree. Additionally to satisfying the usual stability and approximation properties this operator is defined on dual Sobolev spaces. Also it enjoys stability and approximation properties in the corresponding norms leading to optimal rates of convergence. We compare this operator to alternative operators and discuss applications.
This is joint work with Lars Diening and Johannes Storn (Bielefeld University).